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0003042v1	18	[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111 [32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and $$(1,1)$$ -decomposible knots. Preprint, 1999 [34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664 [35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topol- ogy 31 (1992), 271–279 [36] Wu, Y-Q.,: $$\Dot{O}$$ -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331 [37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696 [38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610 LUIGI GRASSELLI, Department of Sciences and Methods for Engineer- ing, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected] MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected]	<p>[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111</p> <p>[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and $$(1,1)$$ -decomposible knots. Preprint, 1999</p> <p>[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664</p> <p>[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topol- ogy 31 (1992), 271–279</p> <p>[36] Wu, Y-Q.,: $$\Dot{O}$$ -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331</p> <p>[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696</p> <p>[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. 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0002044v1	2	# 2.1. The affine fusion ring The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $$X_{r}^{(1)}$$ [23]. Choose any positive integer $$k$$ . Consider the (finite) set $$P_{+}=P_{+}^{k}(X_{r}^{(1)})$$ of level $$k$$ integrable highest weights: where $$\Lambda_{i}$$ denote the fundamental weights, and $$a_{j}^{\vee}$$ are the co-labels, of $$X_{r}^{(1)}$$ (the $$a_{j}^{\vee}$$ will be given for each algebra in §3). We will usually drop the (redundant) component $$\lambda_{0}\Lambda_{0}$$ . Kac- Peterson [24] found a natural representation of the modular group $$\operatorname{SL_{2}}(\mathbb{Z})$$ on the complex space spanned by the affine characters $$\chi_{\mu}$$ , $$\mu\in P_{+}$$ : most significantly, $$\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$$ is sent to the Kac-Peterson matrix $$S$$ with entries An explicit expression for the normalisation constant $$c$$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $$\overline{W}$$ is the (finite) Weyl group of $$X_{r}$$ , and acts on $$P_{+}$$ by fixing $$\Lambda_{0}$$ . The Weyl vector $$\rho$$ equals $$\sum_{i}\Lambda_{i}$$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here. The matrix $$S$$ is symmetric and unitary. One of the weights, $$k\Lambda_{0}$$ , is distinguished and will be denoted ‘ $$0^{\circ}$$ . It is the weight appearing in the denominator of (1.1). A useful fact is that Equation (2.1a) gives us the important where $$\mathrm{ch}_{{\overline{{\lambda}}}}$$ is the Weyl character of the $$X_{r}$$ -module $$L(\overline{{\lambda}})$$ . Together with the Weyl denomi- nator formula, it provides a useful expression for the $$q$$ -dimensions: where the product is over the positive roots $$\alpha\in\overline{{\Delta}}_{+}$$ of $$X_{r}$$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4).	<h1>2.1. The affine fusion ring</h1> <p>The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $$X_{r}^{(1)}$$ [23]. Choose any positive integer $$k$$ . Consider the (finite) set $$P_{+}=P_{+}^{k}(X_{r}^{(1)})$$ of level $$k$$ integrable highest weights:</p> <p>where $$\Lambda_{i}$$ denote the fundamental weights, and $$a_{j}^{\vee}$$ are the co-labels, of $$X_{r}^{(1)}$$ (the $$a_{j}^{\vee}$$ will be given for each algebra in §3). We will usually drop the (redundant) component $$\lambda_{0}\Lambda_{0}$$ . Kac- Peterson [24] found a natural representation of the modular group $$\operatorname{SL_{2}}(\mathbb{Z})$$ on the complex space spanned by the affine characters $$\chi_{\mu}$$ , $$\mu\in P_{+}$$ : most significantly, $$\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$$ is sent to the Kac-Peterson matrix $$S$$ with entries</p> <p>An explicit expression for the normalisation constant $$c$$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $$\overline{W}$$ is the (finite) Weyl group of $$X_{r}$$ , and acts on $$P_{+}$$ by fixing $$\Lambda_{0}$$ . The Weyl vector $$\rho$$ equals $$\sum_{i}\Lambda_{i}$$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.</p> <p>The matrix $$S$$ is symmetric and unitary. One of the weights, $$k\Lambda_{0}$$ , is distinguished and will be denoted ‘ $$0^{\circ}$$ . It is the weight appearing in the denominator of (1.1). A useful fact is that</p> <p>Equation (2.1a) gives us the important</p> <p>where $$\mathrm{ch}_{{\overline{{\lambda}}}}$$ is the Weyl character of the $$X_{r}$$ -module $$L(\overline{{\lambda}})$$ . Together with the Weyl denomi- nator formula, it provides a useful expression for the $$q$$ -dimensions:</p> <p>where the product is over the positive roots $$\alpha\in\overline{{\Delta}}_{+}$$ of $$X_{r}$$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4).</p>	[{"type": "title", "coordinates": [70, 100, 214, 116], "content": "2.1. The affine fusion ring", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [70, 123, 541, 171], "content": "The source of some of the most interesting fusion data are the affine nontwisted\nKac-Moody algebras $$X_{r}^{(1)}$$ [23]. Choose any positive integer $$k$$ . Consider the (finite) set\n$$P_{+}=P_{+}^{k}(X_{r}^{(1)})$$ of level $$k$$ integrable highest weights:", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [174, 187, 436, 226], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [69, 239, 541, 328], "content": "where $$\\Lambda_{i}$$ denote the fundamental weights, and $$a_{j}^{\\vee}$$ are the co-labels, of $$X_{r}^{(1)}$$ (the $$a_{j}^{\\vee}$$ will be\ngiven for each algebra in \u00a73). We will usually drop the (redundant) component $$\\lambda_{0}\\Lambda_{0}$$ . Kac-\nPeterson [24] found a natural representation of the modular group $$\\operatorname{SL_{2}}(\\mathbb{Z})$$ on the complex\nspace spanned by the affine characters $$\\chi_{\\mu}$$ , $$\\mu\\in P_{+}$$ : most significantly, $$\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)$$ is sent\nto the Kac-Peterson matrix $$S$$ with entries", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [175, 338, 435, 378], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 389, 541, 452], "content": "An explicit expression for the normalisation constant $$c$$ is given in e.g. [23, Theorem 13.8(a)].\nThe inner product in (2.1a) is scaled so that the long roots have norm 2. $$\\overline{W}$$ is the (finite)\nWeyl group of $$X_{r}$$ , and acts on $$P_{+}$$ by fixing $$\\Lambda_{0}$$ . The Weyl vector $$\\rho$$ equals $$\\sum_{i}\\Lambda_{i}$$ , and\n\u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 there is P+ here.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 452, 541, 494], "content": "The matrix $$S$$ is symmetric and unitary. One of the weights, $$k\\Lambda_{0}$$ , is distinguished and\nwill be denoted \u2018 $$0^{\\circ}$$ . It is the weight appearing in the denominator of (1.1). A useful fact\nis that", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [233, 497, 378, 511], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [93, 519, 300, 534], "content": "Equation (2.1a) gives us the important", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [218, 549, 393, 580], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [69, 592, 540, 622], "content": "where $$\\mathrm{ch}_{{\\overline{{\\lambda}}}}$$ is the Weyl character of the $$X_{r}$$ -module $$L(\\overline{{\\lambda}})$$ . Together with the Weyl denomi-\nnator formula, it provides a useful expression for the $$q$$ -dimensions:", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [200, 636, 410, 672], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [69, 685, 540, 715], "content": "where the product is over the positive roots $$\\alpha\\in\\overline{{\\Delta}}_{+}$$ of $$X_{r}$$ . Another consequence of (2.1b)\nis the Kac-Walton formula (2.4).", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [72, 102, 213, 118], "content": "2.1. The affine fusion ring", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [93, 125, 541, 141], "content": "The source of some of the most interesting fusion data are the affine nontwisted", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [69, 137, 184, 159], "content": "Kac-Moody algebras ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [184, 140, 208, 154], "content": "X_{r}^{(1)}", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [209, 137, 393, 159], "content": "[23]. Choose any positive integer ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [394, 144, 401, 153], "content": "k", "score": 0.89, "index": 6}, {"type": "text", "coordinates": [401, 137, 542, 159], "content": ". Consider the (finite) set", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [71, 156, 152, 173], "content": "P_{+}=P_{+}^{k}(X_{r}^{(1)})", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [152, 156, 196, 174], "content": " of level ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [196, 160, 203, 169], "content": "k", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [203, 156, 347, 174], "content": " integrable highest weights:", "score": 1.0, "index": 11}, {"type": "interline_equation", "coordinates": [174, 187, 436, 226], "content": "P_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [69, 241, 105, 260], "content": "where ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [105, 246, 117, 257], "content": "\\Lambda_{i}", "score": 0.92, "index": 14}, {"type": "text", "coordinates": [118, 241, 315, 260], "content": " denote the fundamental weights, and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [316, 245, 329, 259], "content": "a_{j}^{\\vee}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [329, 241, 436, 260], "content": " are the co-labels, of ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [436, 241, 460, 256], "content": "X_{r}^{(1)}", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [461, 241, 488, 260], "content": "(the ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [488, 245, 501, 259], "content": "a_{j}^{\\vee}", "score": 0.88, "index": 20}, {"type": "text", "coordinates": [502, 241, 542, 260], "content": " will be", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [70, 258, 480, 274], "content": "given for each algebra in \u00a73). We will usually drop the (redundant) component ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [480, 259, 506, 271], "content": "\\lambda_{0}\\Lambda_{0}", "score": 0.9, "index": 23}, {"type": "text", "coordinates": [507, 258, 540, 274], "content": ". Kac-", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [70, 273, 419, 288], "content": "Peterson [24] found a natural representation of the modular group ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [419, 274, 456, 286], "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "score": 0.9, "index": 26}, {"type": "text", "coordinates": [457, 273, 540, 288], "content": " on the complex", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [70, 293, 277, 311], "content": "space spanned by the affine characters ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [278, 299, 292, 308], "content": "\\chi_{\\mu}", "score": 0.82, "index": 29}, {"type": "text", "coordinates": [293, 293, 299, 311], "content": ", ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [299, 296, 338, 308], "content": "\\mu\\in P_{+}", "score": 0.91, "index": 31}, {"type": "text", "coordinates": [339, 293, 446, 311], "content": ": most significantly,", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [447, 287, 502, 317], "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "score": 0.95, "index": 33}, {"type": "text", "coordinates": [505, 294, 540, 308], "content": "is sent", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [70, 315, 220, 330], "content": "to the Kac-Peterson matrix ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [220, 317, 228, 326], "content": "S", "score": 0.87, "index": 36}, {"type": "text", "coordinates": [229, 315, 295, 330], "content": " with entries", "score": 1.0, "index": 37}, {"type": "interline_equation", "coordinates": [175, 338, 435, 378], "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)\|\\nu+\\rho)}{\\kappa}]\\ .", "score": 0.94, "index": 38}, {"type": "text", "coordinates": [71, 393, 343, 407], "content": "An explicit expression for the normalisation constant ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [344, 397, 349, 403], "content": "c", "score": 0.87, "index": 40}, {"type": "text", "coordinates": [349, 393, 540, 407], "content": " is given in e.g. [23, Theorem 13.8(a)].", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [70, 406, 454, 421], "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [455, 407, 468, 417], "content": "\\overline{W}", "score": 0.92, "index": 43}, {"type": "text", "coordinates": [468, 406, 540, 421], "content": " is the (finite)", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [70, 420, 151, 436], "content": "Weyl group of ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [151, 423, 167, 433], "content": "X_{r}", "score": 0.91, "index": 46}, {"type": "text", "coordinates": [167, 420, 242, 436], "content": ", and acts on ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [242, 423, 258, 434], "content": "P_{+}", "score": 0.91, "index": 48}, {"type": "text", "coordinates": [258, 420, 313, 436], "content": " by fixing ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [314, 423, 328, 434], "content": "\\Lambda_{0}", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [328, 420, 432, 436], "content": ". The Weyl vector ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [432, 426, 439, 434], "content": "\\rho", "score": 0.87, "index": 52}, {"type": "text", "coordinates": [439, 420, 480, 436], "content": " equals", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [480, 422, 512, 435], "content": "\\sum_{i}\\Lambda_{i}", "score": 0.93, "index": 54}, {"type": "text", "coordinates": [512, 420, 541, 436], "content": ", and", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [68, 435, 473, 457], "content": "\u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 there is P+ here.", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [94, 453, 157, 469], "content": "The matrix ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [158, 456, 166, 465], "content": "S", "score": 0.88, "index": 58}, {"type": "text", "coordinates": [166, 453, 409, 469], "content": " is symmetric and unitary. One of the weights, ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [409, 456, 430, 467], "content": "k\\Lambda_{0}", "score": 0.91, "index": 60}, {"type": "text", "coordinates": [430, 453, 541, 469], "content": ", is distinguished and", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [70, 468, 159, 484], "content": "will be denoted \u2018", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [159, 470, 169, 479], "content": "0^{\\circ}", "score": 0.43, "index": 63}, {"type": "text", "coordinates": [169, 468, 541, 484], "content": ". It is the weight appearing in the denominator of (1.1). 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Together with the Weyl denomi-", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [70, 610, 351, 625], "content": "nator formula, it provides a useful expression for the ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [351, 615, 357, 623], "content": "q", "score": 0.86, "index": 77}, {"type": "text", "coordinates": [357, 610, 424, 625], "content": "-dimensions:", "score": 1.0, "index": 78}, {"type": "interline_equation", "coordinates": [200, 636, 410, 672], "content": "{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left\|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left\|\\alpha\\right)/\\kappa\\right)}~,", "score": 0.94, "index": 79}, {"type": "text", "coordinates": [71, 688, 300, 703], "content": "where the product is over the positive roots ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [301, 688, 341, 702], "content": "\\alpha\\in\\overline{{\\Delta}}_{+}", "score": 0.94, "index": 81}, {"type": "text", "coordinates": [341, 688, 357, 703], "content": " of ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [357, 690, 373, 701], "content": "X_{r}", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [373, 688, 540, 703], "content": ". 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The affine fusion ring ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights: ", "page_idx": 2}, {"type": "equation", "text": "$$\nP_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\\vee}$ will be given for each algebra in \u00a73). We will usually drop the (redundant) component $\\lambda_{0}\\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\\operatorname{SL_{2}}(\\mathbb{Z})$ on the complex space spanned by the affine characters $\\chi_{\\mu}$ , $\\mu\\in P_{+}$ : most significantly, $\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)$ is sent to the Kac-Peterson matrix $S$ with entries ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)\|\\nu+\\rho)}{\\kappa}]\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\\Lambda_{0}$ . The Weyl vector $\\rho$ equals $\\sum_{i}\\Lambda_{i}$ , and \u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 there is P+ here. ", "page_idx": 2}, {"type": "text", "text": "The matrix $S$ is symmetric and unitary. One of the weights, $k\\Lambda_{0}$ , is distinguished and will be denoted \u2018 $0^{\\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Equation (2.1a) gives us the important ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\chi_{\\lambda}[\\mu]\\stackrel{\\mathrm{def}}{=}\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\mathrm{ch}_{\\overline{{{\\lambda}}}}(-2\\pi\\mathrm{i}\\frac{\\overline{{{\\mu}}}+\\overline{{{\\rho}}}}{\\kappa})~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\mathrm{ch}_{{\\overline{{\\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\\overline{{\\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions: ", "page_idx": 2}, {"type": "equation", "text": "$$\n{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left\|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left\|\\alpha\\right)/\\kappa\\right)}~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where the product is over the positive roots $\\alpha\\in\\overline{{\\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4). 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We will usually drop the (redundant) component ", "type": "text"}, {"bbox": [480, 259, 506, 271], "score": 0.9, "content": "\\lambda_{0}\\Lambda_{0}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [507, 258, 540, 274], "score": 1.0, "content": ". Kac-", "type": "text"}], "index": 6}, {"bbox": [70, 273, 540, 288], "spans": [{"bbox": [70, 273, 419, 288], "score": 1.0, "content": "Peterson [24] found a natural representation of the modular group ", "type": "text"}, {"bbox": [419, 274, 456, 286], "score": 0.9, "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [457, 273, 540, 288], "score": 1.0, "content": " on the complex", "type": "text"}], "index": 7}, {"bbox": [70, 287, 540, 317], "spans": [{"bbox": [70, 293, 277, 311], "score": 1.0, "content": "space spanned by the affine characters ", "type": "text"}, {"bbox": [278, 299, 292, 308], "score": 0.82, "content": "\\chi_{\\mu}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [293, 293, 299, 311], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [299, 296, 338, 308], "score": 0.91, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [339, 293, 446, 311], "score": 1.0, "content": ": most significantly,", "type": "text"}, {"bbox": [447, 287, 502, 317], "score": 0.95, "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [505, 294, 540, 308], "score": 1.0, "content": "is sent", "type": "text"}], "index": 8}, {"bbox": [70, 315, 295, 330], "spans": [{"bbox": [70, 315, 220, 330], "score": 1.0, "content": "to the Kac-Peterson matrix ", "type": "text"}, {"bbox": [220, 317, 228, 326], "score": 0.87, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [229, 315, 295, 330], "score": 1.0, "content": " with entries", "type": "text"}], "index": 9}], "index": 7}, {"type": "interline_equation", "bbox": [175, 338, 435, 378], "lines": [{"bbox": [175, 338, 435, 378], "spans": [{"bbox": [175, 338, 435, 378], "score": 0.94, "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)\|\\nu+\\rho)}{\\kappa}]\\ .", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 389, 541, 452], "lines": [{"bbox": [71, 393, 540, 407], "spans": [{"bbox": [71, 393, 343, 407], "score": 1.0, "content": "An explicit expression for the normalisation constant ", "type": "text"}, {"bbox": [344, 397, 349, 403], "score": 0.87, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [349, 393, 540, 407], "score": 1.0, "content": " is given in e.g. [23, Theorem 13.8(a)].", "type": "text"}], "index": 11}, {"bbox": [70, 406, 540, 421], "spans": [{"bbox": [70, 406, 454, 421], "score": 1.0, "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. ", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". The Weyl vector ", "type": "text"}, {"bbox": [432, 426, 439, 434], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [439, 420, 480, 436], "score": 1.0, "content": " equals", "type": "text"}, {"bbox": [480, 422, 512, 435], "score": 0.93, "content": "\\sum_{i}\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [512, 420, 541, 436], "score": 1.0, "content": ", and", "type": "text"}], "index": 13}, {"bbox": [68, 435, 473, 457], "spans": [{"bbox": [68, 435, 473, 457], "score": 1.0, "content": "\u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 there is P+ here.", "type": "text"}], "index": 14}], "index": 12.5}, {"type": "text", "bbox": [70, 452, 541, 494], "lines": [{"bbox": [94, 453, 541, 469], "spans": [{"bbox": [94, 453, 157, 469], "score": 1.0, "content": "The matrix ", "type": "text"}, {"bbox": [158, 456, 166, 465], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [166, 453, 409, 469], "score": 1.0, "content": " is symmetric and unitary. One of the weights, ", "type": "text"}, {"bbox": [409, 456, 430, 467], "score": 0.91, "content": "k\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [430, 453, 541, 469], "score": 1.0, "content": ", is distinguished and", "type": "text"}], "index": 15}, {"bbox": [70, 468, 541, 484], "spans": [{"bbox": [70, 468, 159, 484], "score": 1.0, "content": "will be denoted \u2018", "type": "text"}, {"bbox": [159, 470, 169, 479], "score": 0.43, "content": "0^{\\circ}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 468, 541, 484], "score": 1.0, "content": ". It is the weight appearing in the denominator of (1.1). 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", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". 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This is the matrix S appearing in (1.1); \u03a6 there is P+ here.", "type": "text"}], "index": 14}], "index": 12.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [68, 393, 541, 457]}, {"type": "text", "bbox": [70, 452, 541, 494], "lines": [{"bbox": [94, 453, 541, 469], "spans": [{"bbox": [94, 453, 157, 469], "score": 1.0, "content": "The matrix ", "type": "text"}, {"bbox": [158, 456, 166, 465], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [166, 453, 409, 469], "score": 1.0, "content": " is symmetric and unitary. One of the weights, ", "type": "text"}, {"bbox": [409, 456, 430, 467], "score": 0.91, "content": "k\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [430, 453, 541, 469], "score": 1.0, "content": ", is distinguished and", "type": "text"}], "index": 15}, {"bbox": [70, 468, 541, 484], "spans": [{"bbox": [70, 468, 159, 484], "score": 1.0, "content": "will be denoted \u2018", "type": "text"}, {"bbox": [159, 470, 169, 479], "score": 0.43, "content": "0^{\\circ}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 468, 541, 484], "score": 1.0, "content": ". It is the weight appearing in the denominator of (1.1). 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0003042v1	17	[17] Kim, G., Kim, Y., Vesnin, A.: The knot $$5_{2}$$ and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980 [18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000 [19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284 [20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans- form. Groups 2 (1997), 165–182 [21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68 [22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123 [23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167 [24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263 [25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170	<p>[17] Kim, G., Kim, Y., Vesnin, A.: The knot $$5_{2}$$ and cyclically presented groups. J. Korean Math. 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Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170</p>	[{"type": "text", "coordinates": [108, 120, 503, 663], "content": "[17] Kim, G., Kim, Y., Vesnin, A.: The knot $$5_{2}$$ and cyclically presented\ngroups. J. Korean Math. Soc. 35 (1998), 961\u2013980\n[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To\nappear in: Proc. Inter. 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0002044v1	0	# The Automorphisms of Affine Fusion Rings Terry Gannon Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected] # 1. Introduction Verlinde’s formula [33] arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $$g$$ surface with $$t$$ punctures. $$\Phi$$ here is the finite set of ‘primary fields’. The matrix $$S$$ comes from a representation of $$\mathrm{SL_{2}}(\mathbb{Z})$$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $$V_{\star\cdots\star}^{(g)}$$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature. These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $$\mathbb{F}_{p}$$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein. The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients) where $$C$$ is a permutation of $$\Phi$$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $$\pi$$ of $$\Phi$$ obeying are precisely the symmetries of all numbers of the form (1.1a).	<h1>The Automorphisms of Affine Fusion Rings</h1> <p>Terry Gannon</p> <p>Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]</p> <h1>1. Introduction</h1> <p>Verlinde’s formula [33]</p> <p>arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $$g$$ surface with $$t$$ punctures. $$\Phi$$ here is the finite set of ‘primary fields’. The matrix $$S$$ comes from a representation of $$\mathrm{SL_{2}}(\mathbb{Z})$$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $$V_{\star\cdots\star}^{(g)}$$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature.</p> <p>These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $$\mathbb{F}_{p}$$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein.</p> <p>The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients)</p> <p>where $$C$$ is a permutation of $$\Phi$$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $$\pi$$ of $$\Phi$$ obeying</p> <p>are precisely the symmetries of all numbers of the form (1.1a).</p>	[{"type": "title", "coordinates": [140, 66, 471, 87], "content": "The Automorphisms of Affine Fusion Rings", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [260, 115, 351, 129], "content": "Terry Gannon", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [147, 137, 465, 185], "content": "Department of Mathematical Sciences, University of Alberta,\nEdmonton, Canada, T6G 2G1\ne-mail: [email protected]", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [255, 214, 355, 228], "content": "1. Introduction", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [93, 243, 216, 258], "content": "Verlinde\u2019s formula [33]", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [209, 272, 402, 308], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [70, 318, 541, 408], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for\nthe dimensions of conformal blocks on a genus $$g$$ surface with $$t$$ punctures. $$\\Phi$$ here is the\nfinite set of \u2018primary fields\u2019. The matrix $$S$$ comes from a representation of $$\\mathrm{SL_{2}}(\\mathbb{Z})$$ defined\nby the chiral characters of the theory. Contrary to appearances, these numbers $$V_{\\star\\cdots\\star}^{(g)}$$ will\nalways be nonnegative integers. See the excellent bibliography in [6] for references to the\nphysics literature.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 408, 541, 523], "content": "These numbers are remarkable for also arising in several other contexts: for example,\nas dimensions of spaces of generalised theta functions; as certain tensor product coefficients\nin quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $$\\mathbb{F}_{p}$$ ; as\ncertain knot invariants for 3-manifolds; as composition laws of the superselection sectors in\nalgebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator\nalgebras (VOAs); in von Neumann algebras as \u201cConnes\u2019 fusion\u201d; in quantum cohomology;\nand in Lusztig\u2019s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and\nreferences therein.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 523, 541, 552], "content": "The more fundamental of these numbers are those corresponding to a sphere with three\npunctures. It is more convenient to write these in the form (called fusion coefficients)", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [223, 567, 387, 602], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [70, 613, 541, 657], "content": "where $$C$$ is a permutation of $$\\Phi$$ called charge-conjugation and will be defined below. The\nfusion coefficients uniquely determine all other Verlinde dimensions (1.1a). 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Introduction", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [95, 245, 213, 259], "content": "Verlinde\u2019s formula [33]", "score": 1.0, "index": 7}, {"type": "interline_equation", "coordinates": [209, 272, 402, 308], "content": "V_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [70, 322, 541, 338], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [70, 336, 321, 352], "content": "the dimensions of conformal blocks on a genus ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [322, 342, 328, 349], "content": "g", "score": 0.89, "index": 11}, {"type": "text", "coordinates": [328, 336, 400, 352], "content": " surface with ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [401, 339, 405, 347], "content": "t", "score": 0.86, "index": 13}, {"type": "text", "coordinates": [406, 336, 470, 352], "content": " punctures. 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See for example [7,20,19,32,11,10,36,37,26], and", "type": "text"}], "index": 20}, {"bbox": [71, 512, 167, 525], "spans": [{"bbox": [71, 512, 167, 525], "score": 1.0, "content": "references therein.", "type": "text"}], "index": 21}], "index": 17.5}, {"type": "text", "bbox": [70, 523, 541, 552], "lines": [{"bbox": [94, 525, 541, 540], "spans": [{"bbox": [94, 525, 541, 540], "score": 1.0, "content": "The more fundamental of these numbers are those corresponding to a sphere with three", "type": "text"}], "index": 22}, {"bbox": [70, 540, 520, 555], "spans": [{"bbox": [70, 540, 520, 555], "score": 1.0, "content": "punctures. It is more convenient to write these in the form (called fusion coefficients)", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "lines": [{"bbox": [223, 567, 387, 602], "spans": [{"bbox": [223, 567, 387, 602], "score": 0.95, "content": "N_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{}}{S_{0d}}}", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [70, 613, 541, 657], "lines": [{"bbox": [71, 616, 541, 631], "spans": [{"bbox": [71, 616, 106, 631], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 618, 116, 627], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [116, 616, 225, 631], "score": 1.0, "content": " is a permutation of ", "type": "text"}, {"bbox": [226, 618, 234, 627], "score": 0.88, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [235, 616, 541, 631], "score": 1.0, "content": " called charge-conjugation and will be defined below. The", "type": "text"}], "index": 25}, {"bbox": [70, 630, 542, 645], "spans": [{"bbox": [70, 630, 542, 645], "score": 1.0, "content": "fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries", "type": "text"}], "index": 26}, {"bbox": [70, 643, 385, 660], "spans": [{"bbox": [70, 643, 305, 660], "score": 1.0, "content": "of the numbers (1.1b), i.e. the permutations ", "type": "text"}, {"bbox": [305, 649, 313, 655], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [313, 643, 330, 660], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [330, 646, 339, 655], "score": 0.9, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 643, 385, 660], "score": 1.0, "content": " obeying", "type": "text"}], "index": 27}], "index": 26}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "lines": [{"bbox": [266, 673, 344, 689], "spans": [{"bbox": [266, 673, 344, 689], "score": 0.93, "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [70, 700, 399, 715], "lines": [{"bbox": [70, 703, 398, 717], "spans": [{"bbox": [70, 703, 398, 717], "score": 1.0, "content": "are precisely the symmetries of all numbers of the form (1.1a).", "type": "text"}], "index": 29}], "index": 29}], "layout_bboxes": [], "page_idx": 0, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [209, 272, 402, 308], "lines": [{"bbox": [209, 272, 402, 308], "spans": [{"bbox": [209, 272, 402, 308], "score": 0.94, "content": "V_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "lines": [{"bbox": [223, 567, 387, 602], "spans": [{"bbox": [223, 567, 387, 602], "score": 0.95, "content": "N_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{}}{S_{0d}}}", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "lines": [{"bbox": [266, 673, 344, 689], "spans": [{"bbox": [266, 673, 344, 689], "score": 0.93, "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "type": "interline_equation"}], "index": 28}], "index": 28}], "discarded_blocks": [{"type": "discarded", "bbox": [13, 167, 38, 558], "lines": [{"bbox": [13, 168, 37, 556], "spans": [{"bbox": [13, 168, 37, 556], "score": 1.0, "content": "arXiv:math/0002044v1 [math.QA] 7 Feb 2000", "type": "text", "height": 388, "width": 24}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [140, 66, 471, 87], "lines": [{"bbox": [141, 69, 469, 88], "spans": [{"bbox": [141, 69, 469, 88], "score": 1.0, "content": "The Automorphisms of Affine Fusion Rings", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [260, 115, 351, 129], "lines": [{"bbox": [261, 117, 351, 131], "spans": [{"bbox": [261, 117, 351, 131], "score": 1.0, "content": "Terry Gannon", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [261, 117, 351, 131]}, {"type": "text", "bbox": [147, 137, 465, 185], "lines": [{"bbox": [147, 140, 464, 153], "spans": [{"bbox": [147, 140, 464, 153], "score": 1.0, "content": "Department of Mathematical Sciences, University of Alberta,", "type": "text"}], "index": 2}, {"bbox": [227, 154, 386, 167], "spans": [{"bbox": [227, 154, 386, 167], "score": 1.0, "content": "Edmonton, Canada, T6G 2G1", "type": "text"}], "index": 3}, {"bbox": [215, 173, 396, 185], "spans": [{"bbox": [215, 173, 396, 185], "score": 1.0, "content": "e-mail: [email protected]", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [147, 140, 464, 185]}, {"type": "title", "bbox": [255, 214, 355, 228], "lines": [{"bbox": [257, 216, 355, 229], "spans": [{"bbox": [257, 216, 355, 229], "score": 1.0, "content": "1. Introduction", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [93, 243, 216, 258], "lines": [{"bbox": [95, 245, 213, 259], "spans": [{"bbox": [95, 245, 213, 259], "score": 1.0, "content": "Verlinde\u2019s formula [33]", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [95, 245, 213, 259]}, {"type": "interline_equation", "bbox": [209, 272, 402, 308], "lines": [{"bbox": [209, 272, 402, 308], "spans": [{"bbox": [209, 272, 402, 308], "score": 0.94, "content": "V_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 318, 541, 408], "lines": [{"bbox": [70, 322, 541, 338], "spans": [{"bbox": [70, 322, 541, 338], "score": 1.0, "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for", "type": "text"}], "index": 8}, {"bbox": [70, 336, 541, 352], "spans": [{"bbox": [70, 336, 321, 352], "score": 1.0, "content": "the dimensions of conformal blocks on a genus ", "type": "text"}, {"bbox": [322, 342, 328, 349], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [328, 336, 400, 352], "score": 1.0, "content": " surface with ", "type": "text"}, {"bbox": [401, 339, 405, 347], "score": 0.86, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [406, 336, 470, 352], "score": 1.0, "content": " punctures. ", "type": "text"}, {"bbox": [471, 338, 479, 347], "score": 0.89, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [480, 336, 541, 352], "score": 1.0, "content": " here is the", "type": "text"}], "index": 9}, {"bbox": [71, 352, 541, 366], "spans": [{"bbox": [71, 352, 284, 366], "score": 1.0, "content": "finite set of \u2018primary fields\u2019. The matrix ", "type": "text"}, {"bbox": [285, 353, 293, 362], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [293, 352, 461, 366], "score": 1.0, "content": " comes from a representation of ", "type": "text"}, {"bbox": [461, 352, 498, 364], "score": 0.92, "content": "\\mathrm{SL_{2}}(\\mathbb{Z})", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [499, 352, 541, 366], "score": 1.0, "content": " defined", "type": "text"}], "index": 10}, {"bbox": [68, 361, 545, 386], "spans": [{"bbox": [68, 361, 490, 386], "score": 1.0, "content": "by the chiral characters of the theory. Contrary to appearances, these numbers ", "type": "text"}, {"bbox": [491, 365, 516, 380], "score": 0.93, "content": "V_{\\star\\cdots\\star}^{(g)}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [517, 361, 545, 386], "score": 1.0, "content": "will", "type": "text"}], "index": 11}, {"bbox": [71, 382, 541, 396], "spans": [{"bbox": [71, 382, 541, 396], "score": 1.0, "content": "always be nonnegative integers. See the excellent bibliography in [6] for references to the", "type": "text"}], "index": 12}, {"bbox": [70, 397, 165, 410], "spans": [{"bbox": [70, 397, 165, 410], "score": 1.0, "content": "physics literature.", "type": "text"}], "index": 13}], "index": 10.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [68, 322, 545, 410]}, {"type": "text", "bbox": [70, 408, 541, 523], "lines": [{"bbox": [94, 410, 540, 425], "spans": [{"bbox": [94, 410, 540, 425], "score": 1.0, "content": "These numbers are remarkable for also arising in several other contexts: for example,", "type": "text"}], "index": 14}, {"bbox": [71, 426, 541, 439], "spans": [{"bbox": [71, 426, 541, 439], "score": 1.0, "content": "as dimensions of spaces of generalised theta functions; as certain tensor product coefficients", "type": "text"}], "index": 15}, {"bbox": [69, 439, 542, 455], "spans": [{"bbox": [69, 439, 505, 455], "score": 1.0, "content": "in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ", "type": "text"}, {"bbox": [506, 441, 519, 453], "score": 0.91, "content": "\\mathbb{F}_{p}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [519, 439, 542, 455], "score": 1.0, "content": "; as", "type": "text"}], "index": 16}, {"bbox": [70, 454, 540, 467], "spans": [{"bbox": [70, 454, 540, 467], "score": 1.0, "content": "certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in", "type": "text"}], "index": 17}, {"bbox": [72, 469, 540, 482], "spans": [{"bbox": [72, 469, 540, 482], "score": 1.0, "content": "algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator", "type": "text"}], "index": 18}, {"bbox": [70, 482, 541, 497], "spans": [{"bbox": [70, 482, 541, 497], "score": 1.0, "content": "algebras (VOAs); in von Neumann algebras as \u201cConnes\u2019 fusion\u201d; in quantum cohomology;", "type": "text"}], "index": 19}, {"bbox": [70, 496, 541, 511], "spans": [{"bbox": [70, 496, 541, 511], "score": 1.0, "content": "and in Lusztig\u2019s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and", "type": "text"}], "index": 20}, {"bbox": [71, 512, 167, 525], "spans": [{"bbox": [71, 512, 167, 525], "score": 1.0, "content": "references therein.", "type": "text"}], "index": 21}], "index": 17.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [69, 410, 542, 525]}, {"type": "text", "bbox": [70, 523, 541, 552], "lines": [{"bbox": [94, 525, 541, 540], "spans": [{"bbox": [94, 525, 541, 540], "score": 1.0, "content": "The more fundamental of these numbers are those corresponding to a sphere with three", "type": "text"}], "index": 22}, {"bbox": [70, 540, 520, 555], "spans": [{"bbox": [70, 540, 520, 555], "score": 1.0, "content": "punctures. It is more convenient to write these in the form (called fusion coefficients)", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 525, 541, 555]}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "lines": [{"bbox": [223, 567, 387, 602], "spans": [{"bbox": [223, 567, 387, 602], "score": 0.95, "content": "N_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 613, 541, 657], "lines": [{"bbox": [71, 616, 541, 631], "spans": [{"bbox": [71, 616, 106, 631], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 618, 116, 627], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [116, 616, 225, 631], "score": 1.0, "content": " is a permutation of ", "type": "text"}, {"bbox": [226, 618, 234, 627], "score": 0.88, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [235, 616, 541, 631], "score": 1.0, "content": " called charge-conjugation and will be defined below. The", "type": "text"}], "index": 25}, {"bbox": [70, 630, 542, 645], "spans": [{"bbox": [70, 630, 542, 645], "score": 1.0, "content": "fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries", "type": "text"}], "index": 26}, {"bbox": [70, 643, 385, 660], "spans": [{"bbox": [70, 643, 305, 660], "score": 1.0, "content": "of the numbers (1.1b), i.e. the permutations ", "type": "text"}, {"bbox": [305, 649, 313, 655], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [313, 643, 330, 660], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [330, 646, 339, 655], "score": 0.9, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 643, 385, 660], "score": 1.0, "content": " obeying", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 616, 542, 660]}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "lines": [{"bbox": [266, 673, 344, 689], "spans": [{"bbox": [266, 673, 344, 689], "score": 0.93, "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 700, 399, 715], "lines": [{"bbox": [70, 703, 398, 717], "spans": [{"bbox": [70, 703, 398, 717], "score": 1.0, "content": "are precisely the symmetries of all numbers of the form (1.1a).", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 703, 398, 717]}]}
0002044v1	1	The point of introducing the $$N_{a b}^{c}$$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $$\chi_{a}$$ labelled by the $$a\in\Phi$$ ; the multiplication is defined to have structure constants $$N_{a b}^{c}$$ : As an abstract ring, it is not so interesting (the fusion ring over $$\mathbb{C}$$ is isomorphic to $$\mathbb{C}^{\|\|\Phi\|\|}$$ with operations defined component-wise; over $$\mathbb{Q}$$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $$\{\chi_{a}\}$$ , and so proper definitions of isomorphisms etc. must respect that. The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although Moore- Seiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18]. It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries. A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic. Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].	<p>The point of introducing the $$N_{a b}^{c}$$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $$\chi_{a}$$ labelled by the $$a\in\Phi$$ ; the multiplication is defined to have structure constants $$N_{a b}^{c}$$ :</p> <p>As an abstract ring, it is not so interesting (the fusion ring over $$\mathbb{C}$$ is isomorphic to $$\mathbb{C}^{\|\|\Phi\|\|}$$ with operations defined component-wise; over $$\mathbb{Q}$$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $$\{\chi_{a}\}$$ , and so proper definitions of isomorphisms etc. must respect that.</p> <p>The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although Moore- Seiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18].</p> <p>It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries.</p> <p>A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic.</p> <p>Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].</p>	[{"type": "text", "coordinates": [70, 70, 541, 115], "content": "The point of introducing the $$N_{a b}^{c}$$ in (1.1b) is that they define an algebraic structure,\nthe fusion ring. Consider all formal linear combinations of objects $$\\chi_{a}$$ labelled by the\n$$a\\in\\Phi$$ ; the multiplication is defined to have structure constants $$N_{a b}^{c}$$ :", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [258, 130, 353, 160], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 173, 540, 245], "content": "As an abstract ring, it is not so interesting (the fusion ring over $$\\mathbb{C}$$ is isomorphic to $$\\mathbb{C}^{\|\|\\Phi\|\|}$$\nwith operations defined component-wise; over $$\\mathbb{Q}$$ it will be a direct sum of number fields).\nThis is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a\npolynomial ring. Of course it is important in both contexts that we have a preferred basis,\nnamely $$\\{\\chi_{a}\\}$$ , and so proper definitions of isomorphisms etc. must respect that.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [71, 246, 541, 475], "content": "The most important examples of fusion rings are associated to the affine algebras,\nand it is to these that this paper is devoted. Their automorphisms appear explicitly for\ninstance in the classification of modular invariant (i.e. torus) partition functions [17,18],\nand also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For\ninstance, fusion-automorphisms (more generally, -homomorphisms) generate large classes\nof nonnegative integer representations of the fusion-ring, each of which is associated to a\nboundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the\ncoefficient matrix of the torus partition function is a permutation matrix (in which case the\npartition function is called an automorphism invariant), we get a fusion ring automorphism.\nHowever most torus partition functions are not automorphism invariants (although Moore-\nSeiberg assert that there is a sense in which any torus partition function can be interpreted\nas one \u2014 see e.g. [3]), and most fusion ring automorphisms do not correspond to partition\nfunctions. Nevertheless, the two problems are related. The automorphism invariants for\nthe affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1\nbelow) involving q-dimensions will be very useful to us, and conversely the arguments in\nSection 4 of this paper could be used to considerably simplify the proofs of [17,18].", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 476, 541, 518], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and\nin fact the arguments turn out to be rather short. It is remarkable that the answer is so\nsimple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [71, 518, 541, 547], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this\nin section 5 below; as expected most fusion rings with different names are nonisomorphic.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 554, 540, 598], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck\nin Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank\nYi-Zhi Huang for clarifying an issue concerning [8,21].", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [93, 73, 249, 89], "content": "The point of introducing the ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [249, 75, 268, 88], "content": "N_{a b}^{c}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [269, 73, 540, 89], "content": " in (1.1b) is that they define an algebraic structure,", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [70, 87, 439, 104], "content": "the fusion ring. Consider all formal linear combinations of objects ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [440, 93, 453, 101], "content": "\\chi_{a}", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [453, 87, 541, 104], "content": " labelled by the", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [71, 104, 101, 114], "content": "a\\in\\Phi", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [101, 100, 405, 120], "content": "; the multiplication is defined to have structure constants ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [405, 104, 425, 116], "content": "N_{a b}^{c}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [425, 100, 430, 120], "content": ":", "score": 1.0, "index": 10}, {"type": "interline_equation", "coordinates": [258, 130, 353, 160], "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "score": 0.94, "index": 11}, {"type": "text", "coordinates": [69, 174, 412, 192], "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [412, 178, 421, 187], "content": "\\mathbb{C}", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [422, 174, 513, 192], "content": " is isomorphic to ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [513, 176, 539, 187], "content": "\\mathbb{C}^{\|\|\\Phi\|\|}", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [70, 190, 313, 205], "content": "with operations defined component-wise; over ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [314, 192, 324, 203], "content": "\\mathbb{Q}", "score": 0.89, "index": 17}, {"type": "text", "coordinates": [324, 190, 540, 205], "content": " it will be a direct sum of number fields).", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [69, 204, 542, 221], "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [70, 218, 541, 235], "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [70, 234, 113, 249], "content": "namely ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [113, 235, 138, 247], "content": "\\{\\chi_{a}\\}", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [138, 234, 487, 249], "content": ", and so proper definitions of isomorphisms etc. must respect that.", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [94, 247, 541, 263], "content": "The most important examples of fusion rings are associated to the affine algebras,", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [70, 262, 541, 277], "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [70, 277, 541, 291], "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [70, 290, 541, 306], "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [70, 306, 541, 320], "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [71, 319, 542, 334], "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [70, 334, 541, 349], "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [71, 349, 541, 363], "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [70, 363, 540, 377], "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [70, 377, 540, 392], "content": "However most torus partition functions are not automorphism invariants (although Moore-", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [71, 392, 540, 406], "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [71, 407, 540, 420], "content": "as one \u2014 see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 419, 541, 434], "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [71, 435, 541, 448], "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [70, 448, 541, 464], "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [70, 462, 507, 478], "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [94, 477, 541, 492], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [70, 491, 541, 506], "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [70, 506, 498, 520], "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [95, 521, 540, 535], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [70, 536, 538, 550], "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [95, 556, 541, 572], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [70, 571, 540, 585], "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [72, 586, 354, 600], "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "score": 1.0, "index": 47}]	[]	[{"type": "block", "coordinates": [258, 130, 353, 160], "content": "", "caption": ""}, {"type": "inline", "coordinates": [249, 75, 268, 88], "content": "N_{a b}^{c}", "caption": ""}, {"type": "inline", "coordinates": [440, 93, 453, 101], "content": "\\chi_{a}", "caption": ""}, {"type": "inline", "coordinates": [71, 104, 101, 114], "content": "a\\in\\Phi", "caption": ""}, {"type": "inline", "coordinates": [405, 104, 425, 116], "content": "N_{a b}^{c}", "caption": ""}, {"type": "inline", "coordinates": [412, 178, 421, 187], "content": "\\mathbb{C}", "caption": ""}, {"type": "inline", "coordinates": [513, 176, 539, 187], "content": "\\mathbb{C}^{\|\|\\Phi\|\|}", "caption": ""}, {"type": "inline", "coordinates": [314, 192, 324, 203], "content": "\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [113, 235, 138, 247], "content": "\\{\\chi_{a}\\}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "The point of introducing the $N_{a b}^{c}$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $\\chi_{a}$ labelled by the $a\\in\\Phi$ ; the multiplication is defined to have structure constants $N_{a b}^{c}$ : ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "As an abstract ring, it is not so interesting (the fusion ring over $\\mathbb{C}$ is isomorphic to $\\mathbb{C}^{\|\|\\Phi\|\|}$ with operations defined component-wise; over $\\mathbb{Q}$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $\\{\\chi_{a}\\}$ , and so proper definitions of isomorphisms etc. must respect that. ", "page_idx": 1}, {"type": "text", "text": "The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although MooreSeiberg assert that there is a sense in which any torus partition function can be interpreted as one \u2014 see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18]. ", "page_idx": 1}, {"type": "text", "text": "It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries. ", "page_idx": 1}, {"type": "text", "text": "A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic. ", "page_idx": 1}, {"type": "text", "text": "Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21]. 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104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{\|\|\\Phi\|\|}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one \u2014 see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 115], "lines": [{"bbox": [93, 73, 540, 89], "spans": [{"bbox": [93, 73, 249, 89], "score": 1.0, "content": "The point of introducing the ", "type": "text"}, {"bbox": [249, 75, 268, 88], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [269, 73, 540, 89], "score": 1.0, "content": " in (1.1b) is that they define an algebraic structure,", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 73, 541, 120]}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{\|\|\\Phi\|\|}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [69, 174, 542, 249]}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one \u2014 see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 247, 542, 478]}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 477, 541, 520]}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 521, 540, 550]}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 556, 541, 600]}]}
0002044v1	5	root of unity $$q\,=\,\xi_{2m\kappa}$$ for appropriate choice of $$m\in\{1,2,3\}$$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $$L(k,0)$$ . Because of these isomorphisms, we get that the $$N_{\lambda\mu}^{\nu}$$ do indeed lie in $$\mathbb{Z}_{\geq}$$ , for any affine algebra. A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $$2\kappa$$ : $${\cal D}(w.\lambda)\;=\;\operatorname*{det}\left(w\right){\cal D}(\lambda)$$ and $$(w\lambda\|w\lambda)\,\equiv\,(\lambda\|\lambda)$$ (mod $$2\kappa$$ ). The point is that for excep- tional algebras at small levels, the highest weights can often be distinguished by the pair $$(\mathcal{D}(\lambda),(\lambda+\rho\|\lambda+\rho)$$ (mod $$2\kappa$$ )). For example this is true of $$E_{8,5},E_{8,6},F_{4,4}$$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3). An important property obeyed by the matrix $$S$$ for any classical algebra $$X_{r}$$ is rank- level duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $$A_{r}^{(1)}$$ , it is related to the existence of mutually commutative affine subalgbras $$\widehat{\mathrm{sl}(n)}$$ and $$\widehat{\mathrm{sl}}(\widehat{k})$$ in $$\widehat{\mathrm{gl}}(n\widehat{k})$$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $$G(k,N)$$ , to the fusion ring of the algebra $$\mathrm{u}(k)\cong\mathrm{su}(k)\oplus\mathrm{u}(1)$$ at level $$(N-k,N)$$ . Witten used the duality between $$G(k,N)$$ and $$G(N-k,N)$$ to show that the fusion rings of $$\operatorname{u}(k)$$ level $$(N-k,N)$$ and $$\mathrm{u}(N-k)$$ level $$(k,N)$$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $$\textstyle\sum_{i}[x_{i}]\leftrightarrow\sum_{i}[1-x_{i}]$$ of torsion elements of the Bloch group. The Kac-Peterson matrices of $$\widehat{\mathrm{sl}(\ell)}$$ level $$k$$ and $$\widehat{\mathrm{sl}(k)}$$ level $$\ell$$ are related, as are those of $$C_{r,k}$$ and $$C_{k,r}$$ , and $$\widehat{\mathrm{so}(\ell)}$$ level $$k$$ and $$\widehat{\mathrm{so}(k)}$$ level $$\ell$$ . We will need only the symplectic one; the details will be given in §3.3. # 2.2. Symmetries of fusion coefficients Definition 2.1. By an isomorphism between fusion rings $$\mathcal{R}(X_{r,k})$$ and $$\mathcal{R}(Y_{s,m})$$ (with fusion coefficients $$N$$ and $$M$$ respectively) we mean a bijection $$\pi\ :\ P_{+}^{k}(X_{r}^{(1)})\ \to$$ $${\cal P}_{+}^{m}(Y_{s}^{(1)})$$ such that When $$X_{r,k}~=~Y_{s,m}$$ we call $$\pi$$ an automorphism or fusion-symmetry. Call the pair of permutations $$\pi,\pi^{\prime}$$ an $$S$$ -symmetry if The lemma below tells us that fusion- and $$S$$ -symmetries form two isomorphic groups; the former we will label $$\boldsymbol{A}(\boldsymbol{X}_{r,k})$$ . Equation (2.5a) says that the charge-conjugation $$C$$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $$(C,C)$$ is an $$S$$ - symmetry. Because $$N_{0}=I=M_{\tilde{0}}$$ , $$N_{\lambda\mu}^{0}=C_{\lambda\mu}$$ and $$M_{\tilde{\lambda},\tilde{\mu}}^{\tilde{0}}=\widetilde{C}_{\tilde{\lambda},\tilde{\mu}}$$ (we use tilde’s to denote quantities in $$Y_{s}^{(1)}$$ level $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ ), any isomorphism $$\pi$$ must obey $$\pi0=\tilde{0}$$ and $$\widetilde{C}\circ\pi=\pi\circ C$$ . More generally, since $$N_{\lambda}$$ is a permutation matrix of order $${\boldsymbol{n}}$$ iff $$\lambda$$ is a simpl e- current of order $$n$$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get	<p>root of unity $$q\,=\,\xi_{2m\kappa}$$ for appropriate choice of $$m\in\{1,2,3\}$$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $$L(k,0)$$ . Because of these isomorphisms, we get that the $$N_{\lambda\mu}^{\nu}$$ do indeed lie in $$\mathbb{Z}_{\geq}$$ , for any affine algebra.</p> <p>A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $$2\kappa$$ : $${\cal D}(w.\lambda)\;=\;\operatorname*{det}\left(w\right){\cal D}(\lambda)$$ and $$(w\lambda\|w\lambda)\,\equiv\,(\lambda\|\lambda)$$ (mod $$2\kappa$$ ). The point is that for excep- tional algebras at small levels, the highest weights can often be distinguished by the pair $$(\mathcal{D}(\lambda),(\lambda+\rho\|\lambda+\rho)$$ (mod $$2\kappa$$ )). For example this is true of $$E_{8,5},E_{8,6},F_{4,4}$$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3).</p> <p>An important property obeyed by the matrix $$S$$ for any classical algebra $$X_{r}$$ is rank- level duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $$A_{r}^{(1)}$$ , it is related to the existence of mutually commutative affine subalgbras $$\widehat{\mathrm{sl}(n)}$$ and $$\widehat{\mathrm{sl}}(\widehat{k})$$ in $$\widehat{\mathrm{gl}}(n\widehat{k})$$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $$G(k,N)$$ , to the fusion ring of the algebra $$\mathrm{u}(k)\cong\mathrm{su}(k)\oplus\mathrm{u}(1)$$ at level $$(N-k,N)$$ . Witten used the duality between $$G(k,N)$$ and $$G(N-k,N)$$ to show that the fusion rings of $$\operatorname{u}(k)$$ level $$(N-k,N)$$ and $$\mathrm{u}(N-k)$$ level $$(k,N)$$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $$\textstyle\sum_{i}[x_{i}]\leftrightarrow\sum_{i}[1-x_{i}]$$ of torsion elements of the Bloch group.</p> <p>The Kac-Peterson matrices of $$\widehat{\mathrm{sl}(\ell)}$$ level $$k$$ and $$\widehat{\mathrm{sl}(k)}$$ level $$\ell$$ are related, as are those of $$C_{r,k}$$ and $$C_{k,r}$$ , and $$\widehat{\mathrm{so}(\ell)}$$ level $$k$$ and $$\widehat{\mathrm{so}(k)}$$ level $$\ell$$ . We will need only the symplectic one; the details will be given in §3.3.</p> <h1>2.2. Symmetries of fusion coefficients</h1> <p>Definition 2.1. By an isomorphism between fusion rings $$\mathcal{R}(X_{r,k})$$ and $$\mathcal{R}(Y_{s,m})$$ (with fusion coefficients $$N$$ and $$M$$ respectively) we mean a bijection $$\pi\ :\ P_{+}^{k}(X_{r}^{(1)})\ \to$$ $${\cal P}_{+}^{m}(Y_{s}^{(1)})$$ such that</p> <p>When $$X_{r,k}~=~Y_{s,m}$$ we call $$\pi$$ an automorphism or fusion-symmetry. Call the pair of permutations $$\pi,\pi^{\prime}$$ an $$S$$ -symmetry if</p> <p>The lemma below tells us that fusion- and $$S$$ -symmetries form two isomorphic groups; the former we will label $$\boldsymbol{A}(\boldsymbol{X}_{r,k})$$ . Equation (2.5a) says that the charge-conjugation $$C$$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $$(C,C)$$ is an $$S$$ - symmetry. Because $$N_{0}=I=M_{\tilde{0}}$$ , $$N_{\lambda\mu}^{0}=C_{\lambda\mu}$$ and $$M_{\tilde{\lambda},\tilde{\mu}}^{\tilde{0}}=\widetilde{C}_{\tilde{\lambda},\tilde{\mu}}$$ (we use tilde’s to denote quantities in $$Y_{s}^{(1)}$$ level $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ ), any isomorphism $$\pi$$ must obey $$\pi0=\tilde{0}$$ and $$\widetilde{C}\circ\pi=\pi\circ C$$ . More generally, since $$N_{\lambda}$$ is a permutation matrix of order $${\boldsymbol{n}}$$ iff $$\lambda$$ is a simpl e- current of order $$n$$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get</p>	[{"type": "text", "coordinates": [70, 70, 543, 114], "content": "root of unity $$q\\,=\\,\\xi_{2m\\kappa}$$ for appropriate choice of $$m\\in\\{1,2,3\\}$$ . They also arise from the\nHuang-Lepowsky coproduct [21] for the modules of the VOA $$L(k,0)$$ . Because of these\nisomorphisms, we get that the $$N_{\\lambda\\mu}^{\\nu}$$ do indeed lie in $$\\mathbb{Z}_{\\geq}$$ , for any affine algebra.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [71, 115, 545, 200], "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions\nand norms. Q-dimensions vary by at most a sign while norms are constant mod $$2\\kappa$$ :\n$${\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)$$ and $$(w\\lambda\|w\\lambda)\\,\\equiv\\,(\\lambda\|\\lambda)$$ (mod $$2\\kappa$$ ). The point is that for excep-\ntional algebras at small levels, the highest weights can often be distinguished by the pair\n$$(\\mathcal{D}(\\lambda),(\\lambda+\\rho\|\\lambda+\\rho)$$ (mod $$2\\kappa$$ )). For example this is true of $$E_{8,5},E_{8,6},F_{4,4}$$ . This is a useful\nway in practise to use both (2.4) and the Galois action (2.3).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 201, 542, 365], "content": "An important property obeyed by the matrix $$S$$ for any classical algebra $$X_{r}$$ is rank-\nlevel duality. The first appearance of this curious duality seems to be by Frenkel [9],\nbut by now many aspects and generalisations have been explored in the literature. For\n$$A_{r}^{(1)}$$ , it is related to the existence of mutually commutative affine subalgbras $$\\widehat{\\mathrm{sl}(n)}$$ and\n$$\\widehat{\\mathrm{sl}}(\\widehat{k})$$ in $$\\widehat{\\mathrm{gl}}(n\\widehat{k})$$ . Witten has another interpretation of it [37]: he found a natural map (a\nring homomorphism) from the quantum cohomology of the Grassmannian $$G(k,N)$$ , to the\nfusion ring of the algebra $$\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)$$ at level $$(N-k,N)$$ . Witten used the duality\nbetween $$G(k,N)$$ and $$G(N-k,N)$$ to show that the fusion rings of $$\\operatorname{u}(k)$$ level $$(N-k,N)$$\nand $$\\mathrm{u}(N-k)$$ level $$(k,N)$$ should coincide. A considerable generalisation, applying to any\nVOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution\n$$\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]$$ of torsion elements of the Bloch group.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [71, 366, 541, 414], "content": "The Kac-Peterson matrices of $$\\widehat{\\mathrm{sl}(\\ell)}$$ level $$k$$ and $$\\widehat{\\mathrm{sl}(k)}$$ level $$\\ell$$ are related, as are those of\n$$C_{r,k}$$ and $$C_{k,r}$$ , and $$\\widehat{\\mathrm{so}(\\ell)}$$ level $$k$$ and $$\\widehat{\\mathrm{so}(k)}$$ level $$\\ell$$ . We will need only the symplectic one;\nthe details will be given in \u00a73.3.", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [71, 426, 270, 441], "content": "2.2. Symmetries of fusion coefficients", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [71, 446, 541, 493], "content": "Definition 2.1. By an isomorphism between fusion rings $$\\mathcal{R}(X_{r,k})$$ and $$\\mathcal{R}(Y_{s,m})$$\n(with fusion coefficients $$N$$ and $$M$$ respectively) we mean a bijection $$\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to$$\n$${\\cal P}_{+}^{m}(Y_{s}^{(1)})$$ such that", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [198, 496, 413, 513], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [71, 516, 542, 545], "content": "When $$X_{r,k}~=~Y_{s,m}$$ we call $$\\pi$$ an automorphism or fusion-symmetry. Call the pair of\npermutations $$\\pi,\\pi^{\\prime}$$ an $$S$$ -symmetry if", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [225, 556, 387, 570], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [70, 577, 541, 700], "content": "The lemma below tells us that fusion- and $$S$$ -symmetries form two isomorphic groups;\nthe former we will label $$\\boldsymbol{A}(\\boldsymbol{X}_{r,k})$$ . Equation (2.5a) says that the charge-conjugation $$C$$ , and\nmore generally any conjugation, is a fusion-symmetry, while (2.2a) says $$(C,C)$$ is an $$S$$ -\nsymmetry. Because $$N_{0}=I=M_{\\tilde{0}}$$ , $$N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}$$ and $$M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}$$ (we use tilde\u2019s to denote\nquantities in $$Y_{s}^{(1)}$$ level $${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$$ ), any isomorphism $$\\pi$$ must obey $$\\pi0=\\tilde{0}$$ and $$\\widetilde{C}\\circ\\pi=\\pi\\circ C$$ . More\ngenerally, since $$N_{\\lambda}$$ is a permutation matrix of order $${\\boldsymbol{n}}$$ iff $$\\lambda$$ is a simpl e- current of order $$n$$ ,\nwe see that an isomorphism sends simple-currents to simple-currents of equal order. We\nget", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [254, 702, 358, 716], "content": "", "block_type": "interline_equation", "index": 11}]	[{"type": "text", "coordinates": [70, 73, 143, 89], "content": "root of unity ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [144, 75, 192, 87], "content": "q\\,=\\,\\xi_{2m\\kappa}", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [192, 73, 330, 89], "content": " for appropriate choice of ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [331, 75, 399, 87], "content": "m\\in\\{1,2,3\\}", "score": 0.94, "index": 4}, {"type": "text", "coordinates": [399, 73, 541, 89], "content": ". They also arise from the", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 88, 403, 103], "content": "Huang-Lepowsky coproduct [21] for the modules of the VOA ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [404, 89, 439, 101], "content": "L(k,0)", "score": 0.94, "index": 7}, {"type": "text", "coordinates": [440, 88, 541, 103], "content": ". Because of these", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [70, 103, 233, 117], "content": "isomorphisms, we get that the ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [233, 104, 255, 118], "content": "N_{\\lambda\\mu}^{\\nu}", "score": 0.93, "index": 10}, {"type": "text", "coordinates": [255, 103, 343, 117], "content": " do indeed lie in ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [343, 104, 360, 117], "content": "\\mathbb{Z}_{\\geq}", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [360, 103, 481, 117], "content": ", for any affine algebra.", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [94, 116, 548, 133], "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [70, 131, 523, 146], "content": "and norms. Q-dimensions vary by at most a sign while norms are constant mod ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [523, 133, 536, 142], "content": "2\\kappa", "score": 0.81, "index": 16}, {"type": "text", "coordinates": [537, 131, 540, 146], "content": ":", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [71, 146, 195, 159], "content": "{\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [196, 144, 225, 160], "content": " and ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [226, 146, 317, 159], "content": "(w\\lambda\|w\\lambda)\\,\\equiv\\,(\\lambda\|\\lambda)", "score": 0.92, "index": 20}, {"type": "text", "coordinates": [318, 144, 356, 160], "content": " (mod ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [356, 147, 369, 156], "content": "2\\kappa", "score": 0.67, "index": 22}, {"type": "text", "coordinates": [370, 144, 541, 160], "content": "). The point is that for excep-", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [71, 160, 540, 174], "content": "tional algebras at small levels, the highest weights can often be distinguished by the pair", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [71, 175, 172, 188], "content": "(\\mathcal{D}(\\lambda),(\\lambda+\\rho\|\\lambda+\\rho)", "score": 0.85, "index": 25}, {"type": "text", "coordinates": [172, 173, 207, 190], "content": " (mod ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [207, 176, 220, 185], "content": "2\\kappa", "score": 0.65, "index": 27}, {"type": "text", "coordinates": [221, 173, 377, 190], "content": ")). For example this is true of ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [378, 176, 453, 188], "content": "E_{8,5},E_{8,6},F_{4,4}", "score": 0.93, "index": 29}, {"type": "text", "coordinates": [453, 173, 541, 190], "content": ". This is a useful", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [70, 189, 393, 203], "content": "way in practise to use both (2.4) and the Galois action (2.3).", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [94, 202, 338, 218], "content": "An important property obeyed by the matrix ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [338, 205, 347, 214], "content": "S", "score": 0.89, "index": 33}, {"type": "text", "coordinates": [347, 202, 480, 218], "content": " for any classical algebra ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [480, 205, 496, 215], "content": "X_{r}", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [496, 202, 541, 218], "content": " is rank-", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [70, 216, 541, 233], "content": "level duality. The first appearance of this curious duality seems to be by Frenkel [9],", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [70, 232, 542, 246], "content": "but by now many aspects and generalisations have been explored in the literature. For", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [71, 246, 93, 261], "content": "A_{r}^{(1)}", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [93, 244, 489, 265], "content": ", it is related to the existence of mutually commutative affine subalgbras", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [490, 245, 515, 262], "content": "\\widehat{\\mathrm{sl}(n)}", "score": 0.91, "index": 41}, {"type": "text", "coordinates": [515, 244, 542, 265], "content": " and", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [71, 263, 95, 280], "content": "\\widehat{\\mathrm{sl}}(\\widehat{k})", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [96, 265, 114, 281], "content": " in ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [114, 263, 147, 279], "content": "\\widehat{\\mathrm{gl}}(n\\widehat{k})", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [147, 265, 541, 281], "content": ". Witten has another interpretation of it [37]: he found a natural map (a", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [71, 281, 460, 295], "content": "ring homomorphism) from the quantum cohomology of the Grassmannian ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [460, 281, 502, 294], "content": "G(k,N)", "score": 0.93, "index": 48}, {"type": "text", "coordinates": [502, 281, 541, 295], "content": ", to the", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [70, 294, 204, 309], "content": "fusion ring of the algebra ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [205, 296, 306, 308], "content": "\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [306, 294, 350, 309], "content": " at level ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [351, 296, 407, 308], "content": "(N-k,N)", "score": 0.93, "index": 53}, {"type": "text", "coordinates": [407, 294, 541, 309], "content": ". Witten used the duality", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [70, 309, 117, 323], "content": "between ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [118, 310, 159, 322], "content": "G(k,N)", "score": 0.94, "index": 56}, {"type": "text", "coordinates": [160, 309, 186, 323], "content": " and ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [186, 310, 254, 322], "content": "G(N-k,N)", "score": 0.94, "index": 58}, {"type": "text", "coordinates": [254, 309, 426, 323], "content": " to show that the fusion rings of", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [427, 309, 450, 322], "content": "\\operatorname{u}(k)", "score": 0.87, "index": 60}, {"type": "text", "coordinates": [451, 309, 481, 323], "content": " level ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [482, 309, 540, 322], "content": "(N-k,N)", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [70, 322, 94, 338], "content": "and ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [94, 324, 142, 336], "content": "\\mathrm{u}(N-k)", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [143, 322, 173, 338], "content": " level ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [173, 324, 205, 337], "content": "(k,N)", "score": 0.94, "index": 66}, {"type": "text", "coordinates": [206, 322, 541, 338], "content": " should coincide. A considerable generalisation, applying to any", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [70, 337, 542, 353], "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [71, 353, 179, 366], "content": "\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [179, 351, 388, 368], "content": " of torsion elements of the Bloch group.", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [94, 369, 253, 385], "content": "The Kac-Peterson matrices of", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [253, 367, 276, 383], "content": "\\widehat{\\mathrm{sl}(\\ell)}", "score": 0.92, "index": 72}, {"type": "text", "coordinates": [276, 369, 306, 385], "content": " level ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [306, 372, 313, 380], "content": "k", "score": 0.88, "index": 74}, {"type": "text", "coordinates": [313, 369, 339, 385], "content": " and ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [339, 367, 363, 384], "content": "\\widehat{\\mathrm{sl}(k)}", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [364, 369, 393, 385], "content": " level ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [393, 371, 399, 380], "content": "\\ell", "score": 0.77, "index": 78}, {"type": "text", "coordinates": [399, 369, 542, 385], "content": "are related, as are those of", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [71, 389, 93, 402], "content": "C_{r,k}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [93, 387, 120, 404], "content": " and ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [120, 389, 143, 402], "content": "C_{k,r}", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [143, 387, 173, 404], "content": ", and", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [174, 385, 199, 401], "content": "\\widehat{\\mathrm{so}(\\ell)}", "score": 0.89, "index": 84}, {"type": "text", "coordinates": [200, 387, 231, 404], "content": " level ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [231, 389, 238, 398], "content": "k", "score": 0.87, "index": 86}, {"type": "text", "coordinates": [238, 387, 265, 404], "content": " and", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [266, 384, 293, 401], "content": "\\widehat{\\mathrm{so}(k)}", "score": 0.86, "index": 88}, {"type": "text", "coordinates": [293, 387, 324, 404], "content": " level ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [324, 389, 330, 398], "content": "\\ell", "score": 0.86, "index": 90}, {"type": "text", "coordinates": [330, 387, 541, 404], "content": ". We will need only the symplectic one;", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [70, 402, 239, 416], "content": "the details will be given in \u00a73.3.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [72, 429, 269, 442], "content": "2.2. Symmetries of fusion coefficients", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [93, 447, 423, 465], "content": "Definition 2.1. By an isomorphism between fusion rings ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [423, 449, 466, 463], "content": "\\mathcal{R}(X_{r,k})", "score": 0.92, "index": 95}, {"type": "text", "coordinates": [466, 447, 496, 465], "content": " and ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [496, 449, 540, 462], "content": "\\mathcal{R}(Y_{s,m})", "score": 0.9, "index": 97}, {"type": "text", "coordinates": [69, 464, 203, 481], "content": "(with fusion coefficients ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [204, 466, 216, 477], "content": "N", "score": 0.79, "index": 99}, {"type": "text", "coordinates": [216, 464, 245, 481], "content": " and ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [245, 465, 259, 477], "content": "M", "score": 0.64, "index": 101}, {"type": "text", "coordinates": [260, 464, 448, 481], "content": " respectively) we mean a bijection ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [448, 463, 541, 480], "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "score": 0.92, "index": 103}, {"type": "inline_equation", "coordinates": [71, 480, 122, 498], "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "score": 0.94, "index": 104}, {"type": "text", "coordinates": [122, 478, 177, 498], "content": " such that", "score": 1.0, "index": 105}, {"type": "interline_equation", "coordinates": [198, 496, 413, 513], "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "score": 0.9, "index": 106}, {"type": "text", "coordinates": [71, 516, 107, 535], "content": "When ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [107, 518, 176, 532], "content": "X_{r,k}~=~Y_{s,m}", "score": 0.91, "index": 108}, {"type": "text", "coordinates": [176, 516, 224, 535], "content": " we call ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [225, 523, 232, 529], "content": "\\pi", "score": 0.77, "index": 110}, {"type": "text", "coordinates": [233, 516, 544, 535], "content": " an automorphism or fusion-symmetry. Call the pair of", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [71, 533, 142, 547], "content": "permutations ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [143, 533, 167, 546], "content": "\\pi,\\pi^{\\prime}", "score": 0.87, "index": 113}, {"type": "text", "coordinates": [167, 533, 187, 547], "content": " an ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [187, 534, 196, 543], "content": "S", "score": 0.84, "index": 115}, {"type": "text", "coordinates": [196, 533, 267, 547], "content": "-symmetry if", "score": 1.0, "index": 116}, {"type": "interline_equation", "coordinates": [225, 556, 387, 570], "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "score": 0.9, "index": 117}, {"type": "text", "coordinates": [94, 578, 317, 595], "content": "The lemma below tells us that fusion- and ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [317, 582, 325, 591], "content": "S", "score": 0.9, "index": 119}, {"type": "text", "coordinates": [326, 578, 541, 595], "content": "-symmetries form two isomorphic groups;", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [71, 594, 196, 610], "content": "the former we will label ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [196, 595, 238, 608], "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "score": 0.94, "index": 122}, {"type": "text", "coordinates": [239, 594, 504, 610], "content": ". Equation (2.5a) says that the charge-conjugation ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [504, 596, 514, 605], "content": "C", "score": 0.89, "index": 124}, {"type": "text", "coordinates": [514, 594, 541, 610], "content": ", and", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [70, 609, 459, 623], "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [459, 610, 492, 622], "content": "(C,C)", "score": 0.94, "index": 127}, {"type": "text", "coordinates": [492, 609, 527, 623], "content": " is an ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [527, 610, 536, 619], "content": "S", "score": 0.88, "index": 129}, {"type": "text", "coordinates": [536, 609, 541, 623], "content": "-", "score": 1.0, "index": 130}, {"type": "text", "coordinates": [67, 622, 177, 644], "content": "symmetry. Because ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [177, 626, 248, 639], "content": "N_{0}=I=M_{\\tilde{0}}", "score": 0.92, "index": 132}, {"type": "text", "coordinates": [248, 622, 254, 644], "content": ", ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [255, 625, 313, 640], "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "score": 0.94, "index": 134}, {"type": "text", "coordinates": [314, 622, 341, 644], "content": " and ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [341, 623, 407, 642], "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "score": 0.94, "index": 136}, {"type": "text", "coordinates": [408, 622, 542, 644], "content": " (we use tilde\u2019s to denote", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [69, 641, 139, 662], "content": "quantities in ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [140, 643, 162, 657], "content": "Y_{s}^{(1)}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [162, 641, 192, 662], "content": "level ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [192, 650, 203, 656], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "score": 0.85, "index": 141}, {"type": "text", "coordinates": [204, 641, 306, 662], "content": "), any isomorphism ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [306, 650, 313, 656], "content": "\\pi", "score": 0.88, "index": 143}, {"type": "text", "coordinates": [314, 641, 374, 662], "content": " must obey ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [374, 645, 410, 656], "content": "\\pi0=\\tilde{0}", "score": 0.92, "index": 145}, {"type": "text", "coordinates": [410, 641, 435, 662], "content": " and", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [435, 644, 504, 656], "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "score": 0.92, "index": 147}, {"type": "text", "coordinates": [505, 641, 542, 662], "content": ". More", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [70, 659, 153, 675], "content": "generally, since ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [154, 661, 169, 672], "content": "N_{\\lambda}", "score": 0.93, "index": 150}, {"type": "text", "coordinates": [169, 659, 347, 675], "content": " is a permutation matrix of order ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [347, 664, 354, 670], "content": "{\\boldsymbol{n}}", "score": 0.87, "index": 152}, {"type": "text", "coordinates": [355, 659, 372, 675], "content": " iff", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [372, 661, 380, 670], "content": "\\lambda", "score": 0.88, "index": 154}, {"type": "text", "coordinates": [380, 659, 528, 675], "content": " is a simpl e- current of order ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [529, 664, 536, 670], "content": "n", "score": 0.84, "index": 156}, {"type": "text", "coordinates": [537, 659, 541, 675], "content": ",", "score": 1.0, "index": 157}, {"type": "text", "coordinates": [71, 675, 541, 689], "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "score": 1.0, "index": 158}, {"type": "text", "coordinates": [70, 690, 90, 703], "content": "get", "score": 1.0, "index": 159}, {"type": "interline_equation", "coordinates": [254, 702, 358, 716], "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "score": 0.93, "index": 160}]	[]	[{"type": "block", "coordinates": [198, 496, 413, 513], "content": "", "caption": ""}, {"type": "block", "coordinates": [225, 556, 387, 570], "content": "", "caption": ""}, {"type": "block", "coordinates": [254, 702, 358, 716], "content": "", "caption": ""}, {"type": "inline", "coordinates": [144, 75, 192, 87], "content": "q\\,=\\,\\xi_{2m\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [331, 75, 399, 87], "content": "m\\in\\{1,2,3\\}", "caption": ""}, {"type": "inline", "coordinates": [404, 89, 439, 101], "content": "L(k,0)", "caption": ""}, {"type": "inline", "coordinates": [233, 104, 255, 118], "content": "N_{\\lambda\\mu}^{\\nu}", "caption": ""}, {"type": "inline", "coordinates": [343, 104, 360, 117], "content": "\\mathbb{Z}_{\\geq}", "caption": ""}, {"type": "inline", "coordinates": [523, 133, 536, 142], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [71, 146, 195, 159], "content": "{\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)", "caption": ""}, {"type": "inline", "coordinates": [226, 146, 317, 159], "content": "(w\\lambda\|w\\lambda)\\,\\equiv\\,(\\lambda\|\\lambda)", "caption": ""}, {"type": "inline", "coordinates": [356, 147, 369, 156], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [71, 175, 172, 188], "content": "(\\mathcal{D}(\\lambda),(\\lambda+\\rho\|\\lambda+\\rho)", "caption": ""}, {"type": "inline", "coordinates": [207, 176, 220, 185], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [378, 176, 453, 188], "content": "E_{8,5},E_{8,6},F_{4,4}", "caption": ""}, {"type": "inline", "coordinates": [338, 205, 347, 214], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [480, 205, 496, 215], "content": "X_{r}", "caption": ""}, {"type": "inline", "coordinates": [71, 246, 93, 261], "content": "A_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [490, 245, 515, 262], "content": "\\widehat{\\mathrm{sl}(n)}", "caption": ""}, {"type": "inline", "coordinates": [71, 263, 95, 280], "content": "\\widehat{\\mathrm{sl}}(\\widehat{k})", "caption": ""}, {"type": "inline", "coordinates": [114, 263, 147, 279], "content": "\\widehat{\\mathrm{gl}}(n\\widehat{k})", "caption": ""}, {"type": "inline", "coordinates": [460, 281, 502, 294], "content": "G(k,N)", "caption": ""}, {"type": "inline", "coordinates": [205, 296, 306, 308], "content": "\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)", "caption": ""}, {"type": "inline", "coordinates": [351, 296, 407, 308], "content": "(N-k,N)", "caption": ""}, {"type": "inline", "coordinates": [118, 310, 159, 322], "content": "G(k,N)", "caption": ""}, {"type": "inline", 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"content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [71, 389, 93, 402], "content": "C_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [120, 389, 143, 402], "content": "C_{k,r}", "caption": ""}, {"type": "inline", "coordinates": [174, 385, 199, 401], "content": "\\widehat{\\mathrm{so}(\\ell)}", "caption": ""}, {"type": "inline", "coordinates": [231, 389, 238, 398], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [266, 384, 293, 401], "content": "\\widehat{\\mathrm{so}(k)}", "caption": ""}, {"type": "inline", "coordinates": [324, 389, 330, 398], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [423, 449, 466, 463], "content": "\\mathcal{R}(X_{r,k})", "caption": ""}, {"type": "inline", "coordinates": [496, 449, 540, 462], "content": "\\mathcal{R}(Y_{s,m})", "caption": ""}, {"type": "inline", "coordinates": [204, 466, 216, 477], "content": "N", "caption": ""}, {"type": "inline", "coordinates": [245, 465, 259, 477], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [448, 463, 541, 480], "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "caption": ""}, {"type": "inline", "coordinates": [71, 480, 122, 498], "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [107, 518, 176, 532], "content": "X_{r,k}~=~Y_{s,m}", "caption": ""}, {"type": "inline", "coordinates": [225, 523, 232, 529], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [143, 533, 167, 546], "content": "\\pi,\\pi^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [187, 534, 196, 543], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [317, 582, 325, 591], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [196, 595, 238, 608], "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "caption": ""}, {"type": "inline", "coordinates": [504, 596, 514, 605], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [459, 610, 492, 622], "content": "(C,C)", "caption": ""}, {"type": "inline", "coordinates": [527, 610, 536, 619], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [177, 626, 248, 639], "content": "N_{0}=I=M_{\\tilde{0}}", "caption": ""}, {"type": "inline", "coordinates": [255, 625, 313, 640], "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "caption": ""}, {"type": "inline", "coordinates": [341, 623, 407, 642], "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "caption": ""}, {"type": "inline", "coordinates": [140, 643, 162, 657], "content": "Y_{s}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [192, 650, 203, 656], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "caption": ""}, {"type": "inline", "coordinates": [306, 650, 313, 656], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [374, 645, 410, 656], "content": "\\pi0=\\tilde{0}", "caption": ""}, {"type": "inline", "coordinates": [435, 644, 504, 656], "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "caption": ""}, {"type": "inline", "coordinates": [154, 661, 169, 672], "content": "N_{\\lambda}", "caption": ""}, {"type": "inline", "coordinates": [347, 664, 354, 670], "content": "{\\boldsymbol{n}}", "caption": ""}, {"type": "inline", "coordinates": [372, 661, 380, 670], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [529, 664, 536, 670], "content": "n", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 5}, {"type": "text", "text": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $2\\kappa$ : ${\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)$ and $(w\\lambda\|w\\lambda)\\,\\equiv\\,(\\lambda\|\\lambda)$ (mod $2\\kappa$ ). The point is that for exceptional algebras at small levels, the highest weights can often be distinguished by the pair $(\\mathcal{D}(\\lambda),(\\lambda+\\rho\|\\lambda+\\rho)$ (mod $2\\kappa$ )). For example this is true of $E_{8,5},E_{8,6},F_{4,4}$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3). ", "page_idx": 5}, {"type": "text", "text": "An important property obeyed by the matrix $S$ for any classical algebra $X_{r}$ is ranklevel duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $A_{r}^{(1)}$ , it is related to the existence of mutually commutative affine subalgbras $\\widehat{\\mathrm{sl}(n)}$ and $\\widehat{\\mathrm{sl}}(\\widehat{k})$ in $\\widehat{\\mathrm{gl}}(n\\widehat{k})$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $G(k,N)$ , to the fusion ring of the algebra $\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)$ at level $(N-k,N)$ . Witten used the duality between $G(k,N)$ and $G(N-k,N)$ to show that the fusion rings of $\\operatorname{u}(k)$ level $(N-k,N)$ and $\\mathrm{u}(N-k)$ level $(k,N)$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]$ of torsion elements of the Bloch group. ", "page_idx": 5}, {"type": "text", "text": "The Kac-Peterson matrices of $\\widehat{\\mathrm{sl}(\\ell)}$ level $k$ and $\\widehat{\\mathrm{sl}(k)}$ level $\\ell$ are related, as are those of $C_{r,k}$ and $C_{k,r}$ , and $\\widehat{\\mathrm{so}(\\ell)}$ level $k$ and $\\widehat{\\mathrm{so}(k)}$ level $\\ell$ . We will need only the symplectic one; the details will be given in \u00a73.3. ", "page_idx": 5}, {"type": "text", "text": "2.2. Symmetries of fusion coefficients ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "Definition 2.1. By an isomorphism between fusion rings $\\mathcal{R}(X_{r,k})$ and $\\mathcal{R}(Y_{s,m})$ (with fusion coefficients $N$ and $M$ respectively) we mean a bijection $\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to$ ${\\cal P}_{+}^{m}(Y_{s}^{(1)})$ such that ", "page_idx": 5}, {"type": "equation", "text": "$$\nN_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "When $X_{r,k}~=~Y_{s,m}$ we call $\\pi$ an automorphism or fusion-symmetry. Call the pair of permutations $\\pi,\\pi^{\\prime}$ an $S$ -symmetry if ", "page_idx": 5}, {"type": "equation", "text": "$$\nS_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "The lemma below tells us that fusion- and $S$ -symmetries form two isomorphic groups; the former we will label $\\boldsymbol{A}(\\boldsymbol{X}_{r,k})$ . Equation (2.5a) says that the charge-conjugation $C$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $(C,C)$ is an $S$ - symmetry. Because $N_{0}=I=M_{\\tilde{0}}$ , $N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}$ and $M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}$ (we use tilde\u2019s to denote quantities in $Y_{s}^{(1)}$ level ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ ), any isomorphism $\\pi$ must obey $\\pi0=\\tilde{0}$ and $\\widetilde{C}\\circ\\pi=\\pi\\circ C$ . More generally, since $N_{\\lambda}$ is a permutation matrix of order ${\\boldsymbol{n}}$ iff $\\lambda$ is a simpl e- current of order $n$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.\n$$", "text_format": "latex", "page_idx": 5}]	[{"category_id": 1, "poly": [195, 559, 1506, 559, 1506, 1014, 195, 1014], "score": 0.984}, {"category_id": 1, "poly": [195, 1604, 1505, 1604, 1505, 1945, 195, 1945], "score": 0.982}, {"category_id": 1, "poly": [198, 321, 1516, 321, 1516, 558, 198, 558], "score": 0.979}, {"category_id": 1, "poly": [199, 1018, 1504, 1018, 1504, 1152, 199, 1152], "score": 0.967}, {"category_id": 1, "poly": [197, 197, 1509, 197, 1509, 318, 197, 318], "score": 0.965}, {"category_id": 1, "poly": [199, 1240, 1503, 1240, 1503, 1371, 199, 1371], "score": 0.964}, {"category_id": 8, "poly": [624, 1537, 1073, 1537, 1073, 1587, 624, 1587], "score": 0.942}, {"category_id": 8, "poly": [549, 1374, 1147, 1374, 1147, 1425, 549, 1425], "score": 0.941}, {"category_id": 1, "poly": [198, 1435, 1507, 1435, 1507, 1516, 198, 1516], "score": 0.937}, {"category_id": 8, "poly": [702, 1946, 994, 1946, 994, 1991, 702, 1991], "score": 0.918}, {"category_id": 9, "poly": [1429, 1378, 1501, 1378, 1501, 1418, 1429, 1418], "score": 0.886}, {"category_id": 9, "poly": [1412, 1948, 1500, 1948, 1500, 1987, 1412, 1987], "score": 0.885}, {"category_id": 0, "poly": [199, 1184, 750, 1184, 750, 1225, 199, 1225], "score": 0.542}, {"category_id": 0, "poly": [199, 1184, 751, 1184, 751, 1225, 199, 1225], "score": 0.288}, {"category_id": 13, "poly": [328, 862, 444, 862, 444, 896, 328, 896], "score": 0.94, "latex": "G(k,N)"}, {"category_id": 13, "poly": [483, 902, 572, 902, 572, 937, 483, 937], "score": 0.94, "latex": "(k,N)"}, {"category_id": 13, "poly": [949, 1732, 1133, 1732, 1133, 1785, 949, 1785], "score": 0.94, "latex": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}"}, {"category_id": 13, "poly": [1123, 249, 1222, 249, 1222, 283, 1123, 283], "score": 0.94, "latex": "L(k,0)"}, {"category_id": 13, "poly": [709, 1737, 872, 1737, 872, 1779, 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Witten used the duality", "type": "text"}], "index": 15}, {"bbox": [70, 309, 540, 323], "spans": [{"bbox": [70, 309, 117, 323], "score": 1.0, "content": "between ", "type": "text"}, {"bbox": [118, 310, 159, 322], "score": 0.94, "content": "G(k,N)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [160, 309, 186, 323], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 310, 254, 322], "score": 0.94, "content": "G(N-k,N)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [254, 309, 426, 323], "score": 1.0, "content": " to show that the fusion rings of", "type": "text"}, {"bbox": [427, 309, 450, 322], "score": 0.87, "content": "\\operatorname{u}(k)", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [451, 309, 481, 323], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [482, 309, 540, 322], "score": 0.92, "content": "(N-k,N)", "type": "inline_equation", "height": 13, "width": 58}], "index": 16}, {"bbox": [70, 322, 541, 338], "spans": [{"bbox": [70, 322, 94, 338], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 324, 142, 336], "score": 0.94, "content": "\\mathrm{u}(N-k)", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [143, 322, 173, 338], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [173, 324, 205, 337], "score": 0.94, "content": "(k,N)", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [206, 322, 541, 338], "score": 1.0, "content": " should coincide. A considerable generalisation, applying to any", "type": "text"}], "index": 17}, {"bbox": [70, 337, 542, 353], "spans": [{"bbox": [70, 337, 542, 353], "score": 1.0, "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "type": "text"}], "index": 18}, {"bbox": [71, 351, 388, 368], "spans": [{"bbox": [71, 353, 179, 366], "score": 0.93, "content": "\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [179, 351, 388, 368], "score": 1.0, "content": " of torsion elements of the Bloch group.", "type": "text"}], "index": 19}], "index": 14}, {"type": "text", "bbox": [71, 366, 541, 414], "lines": [{"bbox": [94, 367, 542, 385], "spans": [{"bbox": [94, 369, 253, 385], "score": 1.0, "content": "The Kac-Peterson matrices of", "type": "text"}, {"bbox": [253, 367, 276, 383], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(\\ell)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [276, 369, 306, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [306, 372, 313, 380], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [313, 369, 339, 385], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 367, 363, 384], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(k)}", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [364, 369, 393, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [393, 371, 399, 380], "score": 0.77, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [399, 369, 542, 385], "score": 1.0, "content": "are related, as are those of", "type": "text"}], "index": 20}, {"bbox": [71, 384, 541, 404], "spans": [{"bbox": [71, 389, 93, 402], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [93, 387, 120, 404], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [120, 389, 143, 402], "score": 0.92, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [143, 387, 173, 404], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [174, 385, 199, 401], "score": 0.89, "content": "\\widehat{\\mathrm{so}(\\ell)}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [200, 387, 231, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [231, 389, 238, 398], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 387, 265, 404], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [266, 384, 293, 401], "score": 0.86, "content": "\\widehat{\\mathrm{so}(k)}", "type": "inline_equation", "height": 17, "width": 27}, {"bbox": [293, 387, 324, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [324, 389, 330, 398], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [330, 387, 541, 404], "score": 1.0, "content": ". We will need only the symplectic one;", "type": "text"}], "index": 21}, {"bbox": [70, 402, 239, 416], "spans": [{"bbox": [70, 402, 239, 416], "score": 1.0, "content": "the details will be given in \u00a73.3.", "type": "text"}], "index": 22}], "index": 21}, {"type": "title", "bbox": [71, 426, 270, 441], "lines": [{"bbox": [72, 429, 269, 442], "spans": [{"bbox": [72, 429, 269, 442], "score": 1.0, "content": "2.2. Symmetries of fusion coefficients", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [71, 446, 541, 493], "lines": [{"bbox": [93, 447, 540, 465], "spans": [{"bbox": [93, 447, 423, 465], "score": 1.0, "content": "Definition 2.1. By an isomorphism between fusion rings ", "type": "text"}, {"bbox": [423, 449, 466, 463], "score": 0.92, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [466, 447, 496, 465], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 449, 540, 462], "score": 0.9, "content": "\\mathcal{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 44}], "index": 24}, {"bbox": [69, 463, 541, 481], "spans": [{"bbox": [69, 464, 203, 481], "score": 1.0, "content": "(with fusion coefficients ", "type": "text"}, {"bbox": [204, 466, 216, 477], "score": 0.79, "content": "N", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [216, 464, 245, 481], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [245, 465, 259, 477], "score": 0.64, "content": "M", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [260, 464, 448, 481], "score": 1.0, "content": " respectively) we mean a bijection ", "type": "text"}, {"bbox": [448, 463, 541, 480], "score": 0.92, "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "type": "inline_equation", "height": 17, "width": 93}], "index": 25}, {"bbox": [71, 478, 177, 498], "spans": [{"bbox": [71, 480, 122, 498], "score": 0.94, "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 51}, {"bbox": [122, 478, 177, 498], "score": 1.0, "content": " such that", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [71, 516, 542, 545], "lines": [{"bbox": [71, 516, 544, 535], "spans": [{"bbox": [71, 516, 107, 535], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [107, 518, 176, 532], "score": 0.91, "content": "X_{r,k}~=~Y_{s,m}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [176, 516, 224, 535], "score": 1.0, "content": " we call ", "type": "text"}, {"bbox": [225, 523, 232, 529], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [233, 516, 544, 535], "score": 1.0, "content": " an automorphism or fusion-symmetry. Call the pair of", "type": "text"}], "index": 28}, {"bbox": [71, 533, 267, 547], "spans": [{"bbox": [71, 533, 142, 547], "score": 1.0, "content": "permutations ", "type": "text"}, {"bbox": [143, 533, 167, 546], "score": 0.87, "content": "\\pi,\\pi^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [167, 533, 187, 547], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [187, 534, 196, 543], "score": 0.84, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 533, 267, 547], "score": 1.0, "content": "-symmetry if", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [70, 577, 541, 700], "lines": [{"bbox": [94, 578, 541, 595], "spans": [{"bbox": [94, 578, 317, 595], "score": 1.0, "content": "The lemma below tells us that fusion- and ", "type": "text"}, {"bbox": [317, 582, 325, 591], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [326, 578, 541, 595], "score": 1.0, "content": "-symmetries form two isomorphic groups;", "type": "text"}], "index": 31}, {"bbox": [71, 594, 541, 610], "spans": [{"bbox": [71, 594, 196, 610], "score": 1.0, "content": "the former we will label ", "type": "text"}, {"bbox": [196, 595, 238, 608], "score": 0.94, "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [239, 594, 504, 610], "score": 1.0, "content": ". Equation (2.5a) says that the charge-conjugation ", "type": "text"}, {"bbox": [504, 596, 514, 605], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [514, 594, 541, 610], "score": 1.0, "content": ", and", "type": "text"}], "index": 32}, {"bbox": [70, 609, 541, 623], "spans": [{"bbox": [70, 609, 459, 623], "score": 1.0, "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says ", "type": "text"}, {"bbox": [459, 610, 492, 622], "score": 0.94, "content": "(C,C)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [492, 609, 527, 623], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [527, 610, 536, 619], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [536, 609, 541, 623], "score": 1.0, "content": "-", "type": "text"}], "index": 33}, {"bbox": [67, 622, 542, 644], "spans": [{"bbox": [67, 622, 177, 644], "score": 1.0, "content": "symmetry. Because ", "type": "text"}, {"bbox": [177, 626, 248, 639], "score": 0.92, "content": "N_{0}=I=M_{\\tilde{0}}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [248, 622, 254, 644], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [255, 625, 313, 640], "score": 0.94, "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [314, 622, 341, 644], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [341, 623, 407, 642], "score": 0.94, "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "type": "inline_equation", "height": 19, "width": 66}, {"bbox": [408, 622, 542, 644], "score": 1.0, "content": " (we use tilde\u2019s to denote", "type": "text"}], "index": 34}, {"bbox": [69, 641, 542, 662], "spans": [{"bbox": [69, 641, 139, 662], "score": 1.0, "content": "quantities in ", "type": "text"}, {"bbox": [140, 643, 162, 657], "score": 0.92, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [162, 641, 192, 662], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [192, 650, 203, 656], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [204, 641, 306, 662], "score": 1.0, "content": "), any isomorphism ", "type": "text"}, {"bbox": [306, 650, 313, 656], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [314, 641, 374, 662], "score": 1.0, "content": " must obey ", "type": "text"}, {"bbox": [374, 645, 410, 656], "score": 0.92, "content": "\\pi0=\\tilde{0}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [410, 641, 435, 662], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [435, 644, 504, 656], "score": 0.92, "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [505, 641, 542, 662], "score": 1.0, "content": ". More", "type": "text"}], "index": 35}, {"bbox": [70, 659, 541, 675], "spans": [{"bbox": [70, 659, 153, 675], "score": 1.0, "content": "generally, since ", "type": "text"}, {"bbox": [154, 661, 169, 672], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [169, 659, 347, 675], "score": 1.0, "content": " is a permutation matrix of order ", "type": "text"}, {"bbox": [347, 664, 354, 670], "score": 0.87, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 659, 372, 675], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [372, 661, 380, 670], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [380, 659, 528, 675], "score": 1.0, "content": " is a simpl e- current of order ", "type": "text"}, {"bbox": [529, 664, 536, 670], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [537, 659, 541, 675], "score": 1.0, "content": ",", "type": "text"}], "index": 36}, {"bbox": [71, 675, 541, 689], "spans": [{"bbox": [71, 675, 541, 689], "score": 1.0, "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "type": "text"}], "index": 37}, {"bbox": [70, 690, 90, 703], "spans": [{"bbox": [70, 690, 90, 703], "score": 1.0, "content": "get", "type": "text"}], "index": 38}], "index": 34.5}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 543, 114], "lines": [], "index": 1, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 73, 541, 118], "lines_deleted": true}, {"type": "text", "bbox": [71, 115, 545, 200], "lines": [{"bbox": [94, 116, 548, 133], "spans": [{"bbox": [94, 116, 548, 133], "score": 1.0, "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions", "type": "text"}], "index": 3}, {"bbox": [70, 131, 540, 146], "spans": [{"bbox": [70, 131, 523, 146], "score": 1.0, "content": "and norms. Q-dimensions vary by at most a sign while norms are constant mod ", "type": "text"}, {"bbox": [523, 133, 536, 142], "score": 0.81, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [537, 131, 540, 146], "score": 1.0, "content": ":", "type": "text"}], "index": 4}, {"bbox": [71, 144, 541, 160], "spans": [{"bbox": [71, 146, 195, 159], "score": 0.92, "content": "{\\cal D}(w.\\lambda)\\;=\\;\\operatorname{det}\\left(w\\right){\\cal D}(\\lambda)", "type": "inline_equation", "height": 13, "width": 124}, {"bbox": [196, 144, 225, 160], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 146, 317, 159], "score": 0.92, "content": "(w\\lambda\|w\\lambda)\\,\\equiv\\,(\\lambda\|\\lambda)", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [318, 144, 356, 160], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [356, 147, 369, 156], "score": 0.67, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [370, 144, 541, 160], "score": 1.0, "content": "). The point is that for excep-", "type": "text"}], "index": 5}, {"bbox": [71, 160, 540, 174], "spans": [{"bbox": [71, 160, 540, 174], "score": 1.0, "content": "tional algebras at small levels, the highest weights can often be distinguished by the pair", "type": "text"}], "index": 6}, {"bbox": [71, 173, 541, 190], "spans": [{"bbox": [71, 175, 172, 188], "score": 0.85, "content": "(\\mathcal{D}(\\lambda),(\\lambda+\\rho\|\\lambda+\\rho)", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [172, 173, 207, 190], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [207, 176, 220, 185], "score": 0.65, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 173, 377, 190], "score": 1.0, "content": ")). For example this is true of ", "type": "text"}, {"bbox": [378, 176, 453, 188], "score": 0.93, "content": "E_{8,5},E_{8,6},F_{4,4}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [453, 173, 541, 190], "score": 1.0, "content": ". This is a useful", "type": "text"}], "index": 7}, {"bbox": [70, 189, 393, 203], "spans": [{"bbox": [70, 189, 393, 203], "score": 1.0, "content": "way in practise to use both (2.4) and the Galois action (2.3).", "type": "text"}], "index": 8}], "index": 5.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 116, 548, 203]}, {"type": "text", "bbox": [70, 201, 542, 365], "lines": [{"bbox": [94, 202, 541, 218], "spans": [{"bbox": [94, 202, 338, 218], "score": 1.0, "content": "An important property obeyed by the matrix ", "type": "text"}, {"bbox": [338, 205, 347, 214], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 202, 480, 218], "score": 1.0, "content": " for any classical algebra ", "type": "text"}, {"bbox": [480, 205, 496, 215], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [496, 202, 541, 218], "score": 1.0, "content": " is rank-", "type": "text"}], "index": 9}, {"bbox": [70, 216, 541, 233], "spans": [{"bbox": [70, 216, 541, 233], "score": 1.0, "content": "level duality. The first appearance of this curious duality seems to be by Frenkel [9],", "type": "text"}], "index": 10}, {"bbox": [70, 232, 542, 246], "spans": [{"bbox": [70, 232, 542, 246], "score": 1.0, "content": "but by now many aspects and generalisations have been explored in the literature. For", "type": "text"}], "index": 11}, {"bbox": [71, 244, 542, 265], "spans": [{"bbox": [71, 246, 93, 261], "score": 0.92, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [93, 244, 489, 265], "score": 1.0, "content": ", it is related to the existence of mutually commutative affine subalgbras", "type": "text"}, {"bbox": [490, 245, 515, 262], "score": 0.91, "content": "\\widehat{\\mathrm{sl}(n)}", "type": "inline_equation", "height": 17, "width": 25}, {"bbox": [515, 244, 542, 265], "score": 1.0, "content": " and", "type": "text"}], "index": 12}, {"bbox": [71, 263, 541, 281], "spans": [{"bbox": [71, 263, 95, 280], "score": 0.9, "content": "\\widehat{\\mathrm{sl}}(\\widehat{k})", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [96, 265, 114, 281], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [114, 263, 147, 279], "score": 0.91, "content": "\\widehat{\\mathrm{gl}}(n\\widehat{k})", "type": "inline_equation", "height": 16, "width": 33}, {"bbox": [147, 265, 541, 281], "score": 1.0, "content": ". Witten has another interpretation of it [37]: he found a natural map (a", "type": "text"}], "index": 13}, {"bbox": [71, 281, 541, 295], "spans": [{"bbox": [71, 281, 460, 295], "score": 1.0, "content": "ring homomorphism) from the quantum cohomology of the Grassmannian ", "type": "text"}, {"bbox": [460, 281, 502, 294], "score": 0.93, "content": "G(k,N)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [502, 281, 541, 295], "score": 1.0, "content": ", to the", "type": "text"}], "index": 14}, {"bbox": [70, 294, 541, 309], "spans": [{"bbox": [70, 294, 204, 309], "score": 1.0, "content": "fusion ring of the algebra ", "type": "text"}, {"bbox": [205, 296, 306, 308], "score": 0.92, "content": "\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [306, 294, 350, 309], "score": 1.0, "content": " at level ", "type": "text"}, {"bbox": [351, 296, 407, 308], "score": 0.93, "content": "(N-k,N)", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [407, 294, 541, 309], "score": 1.0, "content": ". Witten used the duality", "type": "text"}], "index": 15}, {"bbox": [70, 309, 540, 323], "spans": [{"bbox": [70, 309, 117, 323], "score": 1.0, "content": "between ", "type": "text"}, {"bbox": [118, 310, 159, 322], "score": 0.94, "content": "G(k,N)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [160, 309, 186, 323], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 310, 254, 322], "score": 0.94, "content": "G(N-k,N)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [254, 309, 426, 323], "score": 1.0, "content": " to show that the fusion rings of", "type": "text"}, {"bbox": [427, 309, 450, 322], "score": 0.87, "content": "\\operatorname{u}(k)", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [451, 309, 481, 323], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [482, 309, 540, 322], "score": 0.92, "content": "(N-k,N)", "type": "inline_equation", "height": 13, "width": 58}], "index": 16}, {"bbox": [70, 322, 541, 338], "spans": [{"bbox": [70, 322, 94, 338], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 324, 142, 336], "score": 0.94, "content": "\\mathrm{u}(N-k)", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [143, 322, 173, 338], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [173, 324, 205, 337], "score": 0.94, "content": "(k,N)", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [206, 322, 541, 338], "score": 1.0, "content": " should coincide. A considerable generalisation, applying to any", "type": "text"}], "index": 17}, {"bbox": [70, 337, 542, 353], "spans": [{"bbox": [70, 337, 542, 353], "score": 1.0, "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "type": "text"}], "index": 18}, {"bbox": [71, 351, 388, 368], "spans": [{"bbox": [71, 353, 179, 366], "score": 0.93, "content": "\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [179, 351, 388, 368], "score": 1.0, "content": " of torsion elements of the Bloch group.", "type": "text"}], "index": 19}], "index": 14, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 202, 542, 368]}, {"type": "text", "bbox": [71, 366, 541, 414], "lines": [{"bbox": [94, 367, 542, 385], "spans": [{"bbox": [94, 369, 253, 385], "score": 1.0, "content": "The Kac-Peterson matrices of", "type": "text"}, {"bbox": [253, 367, 276, 383], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(\\ell)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [276, 369, 306, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [306, 372, 313, 380], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [313, 369, 339, 385], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 367, 363, 384], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(k)}", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [364, 369, 393, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [393, 371, 399, 380], "score": 0.77, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [399, 369, 542, 385], "score": 1.0, "content": "are related, as are those of", "type": "text"}], "index": 20}, {"bbox": [71, 384, 541, 404], "spans": [{"bbox": [71, 389, 93, 402], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [93, 387, 120, 404], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [120, 389, 143, 402], "score": 0.92, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [143, 387, 173, 404], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [174, 385, 199, 401], "score": 0.89, "content": "\\widehat{\\mathrm{so}(\\ell)}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [200, 387, 231, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [231, 389, 238, 398], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 387, 265, 404], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [266, 384, 293, 401], "score": 0.86, "content": "\\widehat{\\mathrm{so}(k)}", "type": "inline_equation", "height": 17, "width": 27}, {"bbox": [293, 387, 324, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [324, 389, 330, 398], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [330, 387, 541, 404], "score": 1.0, "content": ". We will need only the symplectic one;", "type": "text"}], "index": 21}, {"bbox": [70, 402, 239, 416], "spans": [{"bbox": [70, 402, 239, 416], "score": 1.0, "content": "the details will be given in \u00a73.3.", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 367, 542, 416]}, {"type": "title", "bbox": [71, 426, 270, 441], "lines": [{"bbox": [72, 429, 269, 442], "spans": [{"bbox": [72, 429, 269, 442], "score": 1.0, "content": "2.2. Symmetries of fusion coefficients", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 446, 541, 493], "lines": [{"bbox": [93, 447, 540, 465], "spans": [{"bbox": [93, 447, 423, 465], "score": 1.0, "content": "Definition 2.1. By an isomorphism between fusion rings ", "type": "text"}, {"bbox": [423, 449, 466, 463], "score": 0.92, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [466, 447, 496, 465], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 449, 540, 462], "score": 0.9, "content": "\\mathcal{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 44}], "index": 24}, {"bbox": [69, 463, 541, 481], "spans": [{"bbox": [69, 464, 203, 481], "score": 1.0, "content": "(with fusion coefficients ", "type": "text"}, {"bbox": [204, 466, 216, 477], "score": 0.79, "content": "N", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [216, 464, 245, 481], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [245, 465, 259, 477], "score": 0.64, "content": "M", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [260, 464, 448, 481], "score": 1.0, "content": " respectively) we mean a bijection ", "type": "text"}, {"bbox": [448, 463, 541, 480], "score": 0.92, "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "type": "inline_equation", "height": 17, "width": 93}], "index": 25}, {"bbox": [71, 478, 177, 498], "spans": [{"bbox": [71, 480, 122, 498], "score": 0.94, "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 51}, {"bbox": [122, 478, 177, 498], "score": 1.0, "content": " such that", "type": "text"}], "index": 26}], "index": 25, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [69, 447, 541, 498]}, {"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 516, 542, 545], "lines": [{"bbox": [71, 516, 544, 535], "spans": [{"bbox": [71, 516, 107, 535], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [107, 518, 176, 532], "score": 0.91, "content": "X_{r,k}~=~Y_{s,m}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [176, 516, 224, 535], "score": 1.0, "content": " we call ", "type": "text"}, {"bbox": [225, 523, 232, 529], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [233, 516, 544, 535], "score": 1.0, "content": " an automorphism or fusion-symmetry. Call the pair of", "type": "text"}], "index": 28}, {"bbox": [71, 533, 267, 547], "spans": [{"bbox": [71, 533, 142, 547], "score": 1.0, "content": "permutations ", "type": "text"}, {"bbox": [143, 533, 167, 546], "score": 0.87, "content": "\\pi,\\pi^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [167, 533, 187, 547], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [187, 534, 196, 543], "score": 0.84, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 533, 267, 547], "score": 1.0, "content": "-symmetry if", "type": "text"}], "index": 29}], "index": 28.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [71, 516, 544, 547]}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 577, 541, 700], "lines": [{"bbox": [94, 578, 541, 595], "spans": [{"bbox": [94, 578, 317, 595], "score": 1.0, "content": "The lemma below tells us that fusion- and ", "type": "text"}, {"bbox": [317, 582, 325, 591], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [326, 578, 541, 595], "score": 1.0, "content": "-symmetries form two isomorphic groups;", "type": "text"}], "index": 31}, {"bbox": [71, 594, 541, 610], "spans": [{"bbox": [71, 594, 196, 610], "score": 1.0, "content": "the former we will label ", "type": "text"}, {"bbox": [196, 595, 238, 608], "score": 0.94, "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [239, 594, 504, 610], "score": 1.0, "content": ". Equation (2.5a) says that the charge-conjugation ", "type": "text"}, {"bbox": [504, 596, 514, 605], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [514, 594, 541, 610], "score": 1.0, "content": ", and", "type": "text"}], "index": 32}, {"bbox": [70, 609, 541, 623], "spans": [{"bbox": [70, 609, 459, 623], "score": 1.0, "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says ", "type": "text"}, {"bbox": [459, 610, 492, 622], "score": 0.94, "content": "(C,C)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [492, 609, 527, 623], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [527, 610, 536, 619], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [536, 609, 541, 623], "score": 1.0, "content": "-", "type": "text"}], "index": 33}, {"bbox": [67, 622, 542, 644], "spans": [{"bbox": [67, 622, 177, 644], "score": 1.0, "content": "symmetry. Because ", "type": "text"}, {"bbox": [177, 626, 248, 639], "score": 0.92, "content": "N_{0}=I=M_{\\tilde{0}}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [248, 622, 254, 644], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [255, 625, 313, 640], "score": 0.94, "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [314, 622, 341, 644], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [341, 623, 407, 642], "score": 0.94, "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "type": "inline_equation", "height": 19, "width": 66}, {"bbox": [408, 622, 542, 644], "score": 1.0, "content": " (we use tilde\u2019s to denote", "type": "text"}], "index": 34}, {"bbox": [69, 641, 542, 662], "spans": [{"bbox": [69, 641, 139, 662], "score": 1.0, "content": "quantities in ", "type": "text"}, {"bbox": [140, 643, 162, 657], "score": 0.92, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [162, 641, 192, 662], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [192, 650, 203, 656], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [204, 641, 306, 662], "score": 1.0, "content": "), any isomorphism ", "type": "text"}, {"bbox": [306, 650, 313, 656], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [314, 641, 374, 662], "score": 1.0, "content": " must obey ", "type": "text"}, {"bbox": [374, 645, 410, 656], "score": 0.92, "content": "\\pi0=\\tilde{0}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [410, 641, 435, 662], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [435, 644, 504, 656], "score": 0.92, "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [505, 641, 542, 662], "score": 1.0, "content": ". More", "type": "text"}], "index": 35}, {"bbox": [70, 659, 541, 675], "spans": [{"bbox": [70, 659, 153, 675], "score": 1.0, "content": "generally, since ", "type": "text"}, {"bbox": [154, 661, 169, 672], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [169, 659, 347, 675], "score": 1.0, "content": " is a permutation matrix of order ", "type": "text"}, {"bbox": [347, 664, 354, 670], "score": 0.87, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 659, 372, 675], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [372, 661, 380, 670], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [380, 659, 528, 675], "score": 1.0, "content": " is a simpl e- current of order ", "type": "text"}, {"bbox": [529, 664, 536, 670], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [537, 659, 541, 675], "score": 1.0, "content": ",", "type": "text"}], "index": 36}, {"bbox": [71, 675, 541, 689], "spans": [{"bbox": [71, 675, 541, 689], "score": 1.0, "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "type": "text"}], "index": 37}, {"bbox": [70, 690, 90, 703], "spans": [{"bbox": [70, 690, 90, 703], "score": 1.0, "content": "get", "type": "text"}], "index": 38}], "index": 34.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [67, 578, 542, 703]}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39, "page_num": "page_5", "page_size": [612.0, 792.0]}]}
0002044v1	4	$$\epsilon_{\ell}(\lambda)/\epsilon_{\ell}^{\prime}(\lambda)=\sigma_{\ell}(c)/c$$ is an unimportant sign independent of $$\lambda$$ . This Galois action will play a fairly important role in this paper. Note that $$\sigma_{-1}=C$$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $$\sigma_{\ell}\circ J=J^{\ell}\circ\sigma_{\ell}$$ . The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $$T_{\lambda\mu}^{\nu}\overset{\mathrm{def}}{=}\mathrm{mult}_{L(\overline{{\lambda}})\otimes L(\overline{{\mu}})}(L(\overline{{\nu}}))$$ in $$X_{r}$$ : where $$w.\gamma\,{\stackrel{\mathrm{def}}{=}}\,w(\gamma+\rho)-\rho$$ and $$W$$ is the affine Weyl group of $$X_{r}^{(1)}$$ (the dependence of $$N_{\lambda\mu}^{\nu}$$ on $$k$$ arises through the action of $$W$$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $$\mathcal{R}(X_{r,k})$$ denote the corresponding fusion ring. A handy consequence of (2.4) that whenever $$k$$ is large enough that $$\lambda+\mu\in P_{+}^{k}(X_{r}^{(1)})$$ (i.e. that $$\begin{array}{r}{\sum_{i=1}^{\tau}a_{i}^{\vee}(\lambda_{i}+\mu_{i})\le k)}\end{array}$$ , then $$N_{\lambda\mu}^{\nu}=T_{\lambda\mu}^{\nu}$$ . It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $$N_{\lambda}$$ , defined by $$(N_{\lambda})_{\mu\nu}=N_{\lambda\mu}^{\nu}$$ . For instance, $$N_{0}=I$$ and, more generally, $$N_{j}$$ is the permutation matrix associated to $$J$$ . The importance of (charge-)conjugation and simple-currents for us is that they respect fusions: for any simple-currents $$J,J^{\prime},j$$ . For example, for $$\mathcal{R}(A_{1,k})$$ we may take $$P_{+}\;=\;\{0,1,\ldots,k\}$$ (the value of $$\lambda_{1}$$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $$j=k$$ is a simple-current corresponding to permutation $$J a=k-a$$ and function $$Q_{j}(a)=a/2$$ . The Galois action sends $$a$$ to the unique weight $$a^{(\ell)}\in P_{+}$$ satisfying $$a^{(\ell)}+1\equiv\pm\ell\left(a+1\right)$$ (mod $$2k+4$$ ), where that sign there equals $$\mathrm{i}^{\ell-1}\epsilon_{\ell}^{\prime}(a)$$ . The fusion coefficients are given by Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $$\mathrm{Ch}(X_{r})/\mathcal{I}_{k}$$ where $$\operatorname{Ch}(X_{r})$$ is the character ring for all finite-dimensional $$X_{r}$$ -modules, and $$\mathcal{I}_{k}$$ is the subspace spanned by the elements $$\mathrm{ch}_{\overline{{\mu}}}-(\operatorname*{det}w)\mathrm{ch}_{\overline{{w}}.\mu}$$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” $$\widetilde{\mathcal{O}}_{k}$$ of Kazhdan-Lusztig’s category of level $$k$$ integrable highest weight $$X_{r}^{(1)}$$ -modules, and t o Gelfand-Kazhdan’s category $$\widetilde{\mathcal{O}}_{q}$$ coming from finite-dimensional modules of the quantum group $$U_{q}X_{r}$$ specialised to the	<p>$$\epsilon_{\ell}(\lambda)/\epsilon_{\ell}^{\prime}(\lambda)=\sigma_{\ell}(c)/c$$ is an unimportant sign independent of $$\lambda$$ . This Galois action will play a fairly important role in this paper. Note that $$\sigma_{-1}=C$$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $$\sigma_{\ell}\circ J=J^{\ell}\circ\sigma_{\ell}$$ .</p> <p>The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $$T_{\lambda\mu}^{\nu}\overset{\mathrm{def}}{=}\mathrm{mult}_{L(\overline{{\lambda}})\otimes L(\overline{{\mu}})}(L(\overline{{\nu}}))$$ in $$X_{r}$$ :</p> <p>where $$w.\gamma\,{\stackrel{\mathrm{def}}{=}}\,w(\gamma+\rho)-\rho$$ and $$W$$ is the affine Weyl group of $$X_{r}^{(1)}$$ (the dependence of $$N_{\lambda\mu}^{\nu}$$ on $$k$$ arises through the action of $$W$$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $$\mathcal{R}(X_{r,k})$$ denote the corresponding fusion ring.</p> <p>A handy consequence of (2.4) that whenever $$k$$ is large enough that $$\lambda+\mu\in P_{+}^{k}(X_{r}^{(1)})$$ (i.e. that $$\begin{array}{r}{\sum_{i=1}^{\tau}a_{i}^{\vee}(\lambda_{i}+\mu_{i})\le k)}\end{array}$$ , then $$N_{\lambda\mu}^{\nu}=T_{\lambda\mu}^{\nu}$$ .</p> <p>It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $$N_{\lambda}$$ , defined by $$(N_{\lambda})_{\mu\nu}=N_{\lambda\mu}^{\nu}$$ . For instance, $$N_{0}=I$$ and, more generally, $$N_{j}$$ is the permutation matrix associated to $$J$$ .</p> <p>The importance of (charge-)conjugation and simple-currents for us is that they respect fusions:</p> <p>for any simple-currents $$J,J^{\prime},j$$ .</p> <p>For example, for $$\mathcal{R}(A_{1,k})$$ we may take $$P_{+}\;=\;\{0,1,\ldots,k\}$$ (the value of $$\lambda_{1}$$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $$j=k$$ is a simple-current corresponding to permutation $$J a=k-a$$ and function $$Q_{j}(a)=a/2$$ . The Galois action sends $$a$$ to the unique weight $$a^{(\ell)}\in P_{+}$$ satisfying $$a^{(\ell)}+1\equiv\pm\ell\left(a+1\right)$$ (mod $$2k+4$$ ), where that sign there equals $$\mathrm{i}^{\ell-1}\epsilon_{\ell}^{\prime}(a)$$ . The fusion coefficients are given by</p> <p>Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $$\mathrm{Ch}(X_{r})/\mathcal{I}_{k}$$ where $$\operatorname{Ch}(X_{r})$$ is the character ring for all finite-dimensional $$X_{r}$$ -modules, and $$\mathcal{I}_{k}$$ is the subspace spanned by the elements $$\mathrm{ch}_{\overline{{\mu}}}-(\operatorname*{det}w)\mathrm{ch}_{\overline{{w}}.\mu}$$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” $$\widetilde{\mathcal{O}}_{k}$$ of Kazhdan-Lusztig’s category of level $$k$$ integrable highest weight $$X_{r}^{(1)}$$ -modules, and t o Gelfand-Kazhdan’s category $$\widetilde{\mathcal{O}}_{q}$$ coming from finite-dimensional modules of the quantum group $$U_{q}X_{r}$$ specialised to the</p>	[{"type": "text", "coordinates": [70, 70, 541, 114], "content": "$$\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c$$ is an unimportant sign independent of $$\\lambda$$ . This Galois action will play\na fairly important role in this paper. Note that $$\\sigma_{-1}=C$$ , so this action can be thought of\nas a generalisation of charge-conjugation. Note also that $$\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [69, 115, 541, 164], "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23\np. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities\n$$T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))$$ in $$X_{r}$$ :", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [239, 178, 372, 209], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [70, 221, 541, 281], "content": "where $$w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho$$ and $$W$$ is the affine Weyl group of $$X_{r}^{(1)}$$ (the dependence of $$N_{\\lambda\\mu}^{\\nu}$$\non $$k$$ arises through the action of $$W$$ ). We shall see shortly that these fusion coefficients,\nnow manifestly integral, are in fact nonnegative. Let $$\\mathcal{R}(X_{r,k})$$ denote the corresponding\nfusion ring.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 282, 541, 313], "content": "A handy consequence of (2.4) that whenever $$k$$ is large enough that $$\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})$$\n(i.e. that $$\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}$$ , then $$N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [70, 313, 541, 354], "content": "It will sometimes be convenient to collect these coefficients in matrix form as the\nfusion matrices $$N_{\\lambda}$$ , defined by $$(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}$$ . For instance, $$N_{0}=I$$ and, more generally,\n$$N_{j}$$ is the permutation matrix associated to $$J$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 355, 540, 383], "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect\nfusions:", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [176, 394, 435, 455], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [70, 463, 235, 477], "content": "for any simple-currents $$J,J^{\\prime},j$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 478, 541, 572], "content": "For example, for $$\\mathcal{R}(A_{1,k})$$ we may take $$P_{+}\\;=\\;\\{0,1,\\ldots,k\\}$$ (the value of $$\\lambda_{1}$$ ), and\nthen the Kac-Peterson matrix is Sab = k2+2 sin(\u03c0 (a+1k)+ (2b+1) ). Charge-conjugation C is\ntrivial here, but $$j=k$$ is a simple-current corresponding to permutation $$J a=k-a$$ and\nfunction $$Q_{j}(a)=a/2$$ . The Galois action sends $$a$$ to the unique weight $$a^{(\\ell)}\\in P_{+}$$ satisfying\n$$a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)$$ (mod $$2k+4$$ ), where that sign there equals $$\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)$$ . The fusion\ncoefficients are given by", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [117, 582, 496, 616], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [70, 624, 541, 716], "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring\n$$\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}$$ where $$\\operatorname{Ch}(X_{r})$$ is the character ring for all finite-dimensional $$X_{r}$$ -modules, and\n$$\\mathcal{I}_{k}$$ is the subspace spanned by the elements $$\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}$$ . Finkelberg [8] proved that\nthis ring is isomorphic to the K-ring of a \u201csub-quotient\u201d $$\\widetilde{\\mathcal{O}}_{k}$$ of Kazhdan-Lusztig\u2019s category\nof level $$k$$ integrable highest weight $$X_{r}^{(1)}$$ -modules, and t o Gelfand-Kazhdan\u2019s category $$\\widetilde{\\mathcal{O}}_{q}$$\ncoming from finite-dimensional modules of the quantum group $$U_{q}X_{r}$$ specialised to the", "block_type": "text", "index": 12}]	[{"type": "inline_equation", "coordinates": [71, 75, 182, 88], "content": "\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c", "score": 0.92, "index": 1}, {"type": "text", "coordinates": [182, 73, 383, 89], "content": " is an unimportant sign independent of ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [383, 75, 391, 84], "content": "\\lambda", "score": 0.89, "index": 3}, {"type": "text", "coordinates": [391, 73, 540, 89], "content": ". This Galois action will play", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [70, 88, 321, 103], "content": "a fairly important role in this paper. Note that ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [321, 90, 367, 101], "content": "\\sigma_{-1}=C", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [367, 88, 542, 103], "content": ", so this action can be thought of", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [70, 101, 370, 118], "content": "as a generalisation of charge-conjugation. Note also that ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [371, 102, 453, 115], "content": "\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [453, 101, 458, 118], "content": ".", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [94, 115, 541, 132], "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [69, 132, 541, 146], "content": "p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [71, 146, 215, 166], "content": "T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [215, 148, 232, 168], "content": " in ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [232, 151, 248, 162], "content": "X_{r}", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [248, 148, 255, 168], "content": ":", "score": 1.0, "index": 16}, {"type": "interline_equation", "coordinates": [239, 178, 372, 209], "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "score": 0.95, "index": 17}, {"type": "text", "coordinates": [66, 218, 105, 247], "content": "where ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [105, 223, 206, 240], "content": "w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [207, 218, 231, 247], "content": " and ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [232, 227, 245, 237], "content": "W", "score": 0.81, "index": 21}, {"type": "text", "coordinates": [245, 218, 388, 247], "content": " is the affine Weyl group of ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [388, 223, 412, 238], "content": "X_{r}^{(1)}", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [412, 218, 517, 247], "content": "(the dependence of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [518, 228, 539, 242], "content": "N_{\\lambda\\mu}^{\\nu}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [69, 239, 88, 256], "content": "on ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [88, 242, 95, 251], "content": "k", "score": 0.84, "index": 27}, {"type": "text", "coordinates": [96, 239, 249, 256], "content": " arises through the action of ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [249, 243, 262, 252], "content": "W", "score": 0.85, "index": 29}, {"type": "text", "coordinates": [263, 239, 541, 256], "content": "). We shall see shortly that these fusion coefficients,", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [69, 254, 358, 270], "content": "now manifestly integral, are in fact nonnegative. Let ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [359, 256, 401, 268], "content": "\\mathcal{R}(X_{r,k})", "score": 0.95, "index": 32}, {"type": "text", "coordinates": [402, 254, 541, 270], "content": " denote the corresponding", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [70, 268, 131, 285], "content": "fusion ring.", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [93, 283, 330, 301], "content": "A handy consequence of (2.4) that whenever ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [330, 287, 337, 296], "content": "k", "score": 0.91, "index": 36}, {"type": "text", "coordinates": [338, 283, 448, 301], "content": " is large enough that ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [448, 283, 539, 300], "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "score": 0.92, "index": 38}, {"type": "text", "coordinates": [69, 297, 120, 318], "content": "(i.e. that", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [121, 300, 237, 314], "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "score": 0.91, "index": 40}, {"type": "text", "coordinates": [237, 297, 272, 318], "content": ", then ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [273, 302, 330, 315], "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "score": 0.94, "index": 42}, {"type": "text", "coordinates": [330, 297, 335, 318], "content": ".", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [93, 313, 540, 327], "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [69, 327, 154, 343], "content": "fusion matrices ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [155, 330, 171, 340], "content": "N_{\\lambda}", "score": 0.91, "index": 46}, {"type": "text", "coordinates": [171, 327, 236, 343], "content": ", defined by ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [236, 329, 311, 344], "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "score": 0.9, "index": 48}, {"type": "text", "coordinates": [312, 327, 392, 343], "content": ". For instance, ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [392, 330, 430, 341], "content": "N_{0}=I", "score": 0.94, "index": 50}, {"type": "text", "coordinates": [430, 327, 541, 343], "content": " and, more generally,", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [71, 344, 86, 357], "content": "N_{j}", "score": 0.91, "index": 52}, {"type": "text", "coordinates": [86, 343, 301, 356], "content": " is the permutation matrix associated to ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [302, 344, 309, 353], "content": "J", "score": 0.82, "index": 54}, {"type": "text", "coordinates": [310, 343, 315, 356], "content": ".", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [93, 355, 542, 374], "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [70, 371, 112, 387], "content": "fusions:", "score": 1.0, "index": 57}, {"type": "interline_equation", "coordinates": [176, 394, 435, 455], "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "score": 0.93, "index": 58}, {"type": "text", "coordinates": [70, 465, 195, 479], "content": "for any simple-currents ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [195, 465, 230, 478], "content": "J,J^{\\prime},j", "score": 0.92, "index": 60}, {"type": "text", "coordinates": [230, 465, 233, 479], "content": ".", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [93, 479, 190, 495], "content": "For example, for ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [190, 480, 233, 494], "content": "\\mathcal{R}(A_{1,k})", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [234, 479, 312, 495], "content": " we may take ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [313, 479, 414, 493], "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "score": 0.89, "index": 65}, {"type": "text", "coordinates": [414, 479, 493, 495], "content": " (the value of ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [493, 482, 506, 492], "content": "\\lambda_{1}", "score": 0.85, "index": 67}, {"type": "text", "coordinates": [506, 479, 542, 495], "content": "), and", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [64, 489, 548, 523], "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(\u03c0 (a+1k)+ (2b+1) ). Charge-conjugation C is", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [70, 515, 159, 530], "content": "trivial here, but ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [159, 516, 189, 529], "content": "j=k", "score": 0.9, "index": 71}, {"type": "text", "coordinates": [190, 515, 456, 530], "content": " is a simple-current corresponding to permutation ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [456, 517, 516, 527], "content": "J a=k-a", "score": 0.91, "index": 73}, {"type": "text", "coordinates": [516, 515, 542, 530], "content": " and", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [69, 527, 117, 547], "content": "function ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [118, 531, 182, 544], "content": "Q_{j}(a)=a/2", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [182, 527, 317, 547], "content": ". The Galois action sends ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [317, 533, 325, 541], "content": "a", "score": 0.69, "index": 78}, {"type": "text", "coordinates": [325, 527, 437, 547], "content": " to the unique weight ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [438, 529, 486, 543], "content": "a^{(\\ell)}\\in P_{+}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [487, 527, 542, 547], "content": " satisfying", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [71, 544, 185, 558], "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [185, 542, 221, 561], "content": " (mod ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [221, 544, 257, 557], "content": "2k+4", "score": 0.46, "index": 84}, {"type": "text", "coordinates": [258, 542, 425, 561], "content": "), where that sign there equals ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [425, 545, 471, 558], "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "score": 0.93, "index": 86}, {"type": "text", "coordinates": [472, 542, 542, 561], "content": ". The fusion", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [71, 559, 196, 574], "content": "coefficients are given by", "score": 1.0, "index": 88}, {"type": "interline_equation", "coordinates": [117, 582, 496, 616], "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}\|a\\!-\\!b\|\\leq c\\leq\\operatorname{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "score": 0.84, "index": 89}, {"type": "text", "coordinates": [95, 627, 541, 642], "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [71, 642, 131, 655], "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "score": 0.91, "index": 91}, {"type": "text", "coordinates": [131, 640, 168, 656], "content": " where ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [169, 642, 209, 655], "content": "\\operatorname{Ch}(X_{r})", "score": 0.9, "index": 93}, {"type": "text", "coordinates": [209, 640, 450, 656], "content": " is the character ring for all finite-dimensional ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [451, 643, 466, 654], "content": "X_{r}", "score": 0.92, "index": 95}, {"type": "text", "coordinates": [466, 640, 542, 656], "content": "-modules, and", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [71, 658, 85, 668], "content": "\\mathcal{I}_{k}", "score": 0.91, "index": 97}, {"type": "text", "coordinates": [86, 656, 300, 671], "content": " is the subspace spanned by the elements ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [300, 657, 397, 670], "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "score": 0.92, "index": 99}, {"type": "text", "coordinates": [397, 656, 541, 671], "content": ". 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"text": "$\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c$ is an unimportant sign independent of $\\lambda$ . This Galois action will play a fairly important role in this paper. Note that $\\sigma_{-1}=C$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}$ . ", "page_idx": 4}, {"type": "text", "text": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))$ in $X_{r}$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\nN_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho$ and $W$ is the affine Weyl group of $X_{r}^{(1)}$ (the dependence of $N_{\\lambda\\mu}^{\\nu}$ on $k$ arises through the action of $W$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $\\mathcal{R}(X_{r,k})$ denote the corresponding fusion ring. ", "page_idx": 4}, {"type": "text", "text": "A handy consequence of (2.4) that whenever $k$ is large enough that $\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})$ (i.e. that $\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}$ , then $N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}$ . ", "page_idx": 4}, {"type": "text", "text": "It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $N_{\\lambda}$ , defined by $(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}$ . For instance, $N_{0}=I$ and, more generally, $N_{j}$ is the permutation matrix associated to $J$ . ", "page_idx": 4}, {"type": "text", "text": "The importance of (charge-)conjugation and simple-currents for us is that they respect fusions: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "for any simple-currents $J,J^{\\prime},j$ . ", "page_idx": 4}, {"type": "text", "text": "For example, for $\\mathcal{R}(A_{1,k})$ we may take $P_{+}\\;=\\;\\{0,1,\\ldots,k\\}$ (the value of $\\lambda_{1}$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(\u03c0 (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $j=k$ is a simple-current corresponding to permutation $J a=k-a$ and function $Q_{j}(a)=a/2$ . The Galois action sends $a$ to the unique weight $a^{(\\ell)}\\in P_{+}$ satisfying $a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)$ (mod $2k+4$ ), where that sign there equals $\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)$ . The fusion coefficients are given by ", "page_idx": 4}, {"type": "equation", "text": "$$\nN_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}\|a\\!-\\!b\|\\leq c\\leq\\operatorname{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}$ where $\\operatorname{Ch}(X_{r})$ is the character ring for all finite-dimensional $X_{r}$ -modules, and $\\mathcal{I}_{k}$ is the subspace spanned by the elements $\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a \u201csub-quotient\u201d $\\widetilde{\\mathcal{O}}_{k}$ of Kazhdan-Lusztig\u2019s category of level $k$ integrable highest weight $X_{r}^{(1)}$ -modules, and t o Gelfand-Kazhdan\u2019s category $\\widetilde{\\mathcal{O}}_{q}$ coming from finite-dimensional modules of the quantum group $U_{q}X_{r}$ specialised to the root of unity $q\\,=\\,\\xi_{2m\\kappa}$ for appropriate choice of $m\\in\\{1,2,3\\}$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $L(k,0)$ . Because of these isomorphisms, we get that the $N_{\\lambda\\mu}^{\\nu}$ do indeed lie in $\\mathbb{Z}_{\\geq}$ , for any affine algebra. 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We shall see shortly that these fusion coefficients,", "type": "text"}], "index": 8}, {"bbox": [69, 254, 541, 270], "spans": [{"bbox": [69, 254, 358, 270], "score": 1.0, "content": "now manifestly integral, are in fact nonnegative. Let ", "type": "text"}, {"bbox": [359, 256, 401, 268], "score": 0.95, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [402, 254, 541, 270], "score": 1.0, "content": " denote the corresponding", "type": "text"}], "index": 9}, {"bbox": [70, 268, 131, 285], "spans": [{"bbox": [70, 268, 131, 285], "score": 1.0, "content": "fusion ring.", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "text", "bbox": [70, 282, 541, 313], "lines": [{"bbox": [93, 283, 539, 301], "spans": [{"bbox": [93, 283, 330, 301], "score": 1.0, "content": "A handy consequence of (2.4) that whenever ", "type": "text"}, {"bbox": [330, 287, 337, 296], "score": 0.91, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [338, 283, 448, 301], "score": 1.0, "content": " is large enough that ", "type": "text"}, {"bbox": [448, 283, 539, 300], "score": 0.92, "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 91}], "index": 11}, {"bbox": [69, 297, 335, 318], "spans": [{"bbox": [69, 297, 120, 318], "score": 1.0, "content": "(i.e. that", "type": "text"}, {"bbox": [121, 300, 237, 314], "score": 0.91, "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "type": "inline_equation", "height": 14, "width": 116}, {"bbox": [237, 297, 272, 318], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [273, 302, 330, 315], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [330, 297, 335, 318], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [70, 313, 541, 354], "lines": [{"bbox": [93, 313, 540, 327], "spans": [{"bbox": [93, 313, 540, 327], "score": 1.0, "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "type": "text"}], "index": 13}, {"bbox": [69, 327, 541, 344], "spans": [{"bbox": [69, 327, 154, 343], "score": 1.0, "content": "fusion matrices ", "type": "text"}, {"bbox": [155, 330, 171, 340], "score": 0.91, "content": "N_{\\lambda}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [171, 327, 236, 343], "score": 1.0, "content": ", defined by ", "type": "text"}, {"bbox": [236, 329, 311, 344], "score": 0.9, "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 15, "width": 75}, {"bbox": [312, 327, 392, 343], "score": 1.0, "content": ". For instance, ", "type": "text"}, {"bbox": [392, 330, 430, 341], "score": 0.94, "content": "N_{0}=I", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [430, 327, 541, 343], "score": 1.0, "content": " and, more generally,", "type": "text"}], "index": 14}, {"bbox": [71, 343, 315, 357], "spans": [{"bbox": [71, 344, 86, 357], "score": 0.91, "content": "N_{j}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [86, 343, 301, 356], "score": 1.0, "content": " is the permutation matrix associated to ", "type": "text"}, {"bbox": [302, 344, 309, 353], "score": 0.82, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [310, 343, 315, 356], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [70, 355, 540, 383], "lines": [{"bbox": [93, 355, 542, 374], "spans": [{"bbox": [93, 355, 542, 374], "score": 1.0, "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "type": "text"}], "index": 16}, {"bbox": [70, 371, 112, 387], "spans": [{"bbox": [70, 371, 112, 387], "score": 1.0, "content": "fusions:", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [70, 463, 235, 477], "lines": [{"bbox": [70, 465, 233, 479], "spans": [{"bbox": [70, 465, 195, 479], "score": 1.0, "content": "for any simple-currents ", "type": "text"}, {"bbox": [195, 465, 230, 478], "score": 0.92, "content": "J,J^{\\prime},j", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [230, 465, 233, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 478, 541, 572], "lines": [{"bbox": [93, 479, 542, 495], "spans": [{"bbox": [93, 479, 190, 495], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [190, 480, 233, 494], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [234, 479, 312, 495], "score": 1.0, "content": " we may take ", "type": "text"}, {"bbox": [313, 479, 414, 493], "score": 0.89, "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [414, 479, 493, 495], "score": 1.0, "content": " (the value of ", "type": "text"}, {"bbox": [493, 482, 506, 492], "score": 0.85, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [506, 479, 542, 495], "score": 1.0, "content": "), and", "type": "text"}], "index": 20}, {"bbox": [64, 489, 548, 523], "spans": [{"bbox": [64, 489, 548, 523], "score": 1.0, "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(\u03c0 (a+1k)+ (2b+1) ). Charge-conjugation C is", "type": "text"}], "index": 21}, {"bbox": [70, 515, 542, 530], "spans": [{"bbox": [70, 515, 159, 530], "score": 1.0, "content": "trivial here, but ", "type": "text"}, {"bbox": [159, 516, 189, 529], "score": 0.9, "content": "j=k", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [190, 515, 456, 530], "score": 1.0, "content": " is a simple-current corresponding to permutation ", "type": "text"}, {"bbox": [456, 517, 516, 527], "score": 0.91, "content": "J a=k-a", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [516, 515, 542, 530], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [69, 527, 542, 547], "spans": [{"bbox": [69, 527, 117, 547], "score": 1.0, "content": "function ", "type": "text"}, {"bbox": [118, 531, 182, 544], "score": 0.92, "content": "Q_{j}(a)=a/2", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [182, 527, 317, 547], "score": 1.0, "content": ". The Galois action sends ", "type": "text"}, {"bbox": [317, 533, 325, 541], "score": 0.69, "content": "a", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [325, 527, 437, 547], "score": 1.0, "content": " to the unique weight ", "type": "text"}, {"bbox": [438, 529, 486, 543], "score": 0.92, "content": "a^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 48}, {"bbox": [487, 527, 542, 547], "score": 1.0, "content": " satisfying", "type": "text"}], "index": 23}, {"bbox": [71, 542, 542, 561], "spans": [{"bbox": [71, 544, 185, 558], "score": 0.91, "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [185, 542, 221, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [221, 544, 257, 557], "score": 0.46, "content": "2k+4", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [258, 542, 425, 561], "score": 1.0, "content": "), where that sign there equals ", "type": "text"}, {"bbox": [425, 545, 471, 558], "score": 0.93, "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [472, 542, 542, 561], "score": 1.0, "content": ". The fusion", "type": "text"}], "index": 24}, {"bbox": [71, 559, 196, 574], "spans": [{"bbox": [71, 559, 196, 574], "score": 1.0, "content": "coefficients are given by", "type": "text"}], "index": 25}], "index": 22.5}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}\|a\\!-\\!b\|\\leq c\\leq\\operatorname{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [70, 624, 541, 716], "lines": [{"bbox": [95, 627, 541, 642], "spans": [{"bbox": [95, 627, 541, 642], "score": 1.0, "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "type": "text"}], "index": 27}, {"bbox": [71, 640, 542, 656], "spans": [{"bbox": [71, 642, 131, 655], "score": 0.91, "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 640, 168, 656], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [169, 642, 209, 655], "score": 0.9, "content": "\\operatorname{Ch}(X_{r})", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [209, 640, 450, 656], "score": 1.0, "content": " is the character ring for all finite-dimensional ", "type": "text"}, {"bbox": [451, 643, 466, 654], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [466, 640, 542, 656], "score": 1.0, "content": "-modules, and", "type": "text"}], "index": 28}, {"bbox": [71, 656, 541, 671], "spans": [{"bbox": [71, 658, 85, 668], "score": 0.91, "content": "\\mathcal{I}_{k}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [86, 656, 300, 671], "score": 1.0, "content": " is the subspace spanned by the elements ", "type": "text"}, {"bbox": [300, 657, 397, 670], "score": 0.92, "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [397, 656, 541, 671], "score": 1.0, "content": ". Finkelberg [8] proved that", "type": "text"}], "index": 29}, {"bbox": [70, 671, 540, 687], "spans": [{"bbox": [70, 672, 363, 687], "score": 1.0, "content": "this ring is isomorphic to the K-ring of a \u201csub-quotient\u201d", "type": "text"}, {"bbox": [363, 671, 379, 685], "score": 0.91, "content": "\\widetilde{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [379, 672, 540, 687], "score": 1.0, "content": " of Kazhdan-Lusztig\u2019s category", "type": "text"}], "index": 30}, {"bbox": [69, 684, 539, 704], "spans": [{"bbox": [69, 684, 111, 704], "score": 1.0, "content": "of level ", "type": "text"}, {"bbox": [112, 690, 119, 699], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [119, 684, 257, 704], "score": 1.0, "content": " integrable highest weight ", "type": "text"}, {"bbox": [257, 686, 281, 700], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [281, 684, 524, 704], "score": 1.0, "content": "-modules, and t o Gelfand-Kazhdan\u2019s category", "type": "text"}, {"bbox": [525, 687, 539, 702], "score": 0.92, "content": "\\widetilde{\\mathcal{O}}_{q}", "type": "inline_equation", "height": 15, "width": 14}], "index": 31}, {"bbox": [71, 703, 541, 717], "spans": [{"bbox": [71, 703, 412, 717], "score": 1.0, "content": "coming from finite-dimensional modules of the quantum group ", "type": "text"}, {"bbox": [413, 704, 441, 717], "score": 0.93, "content": "U_{q}X_{r}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [442, 703, 541, 717], "score": 1.0, "content": " specialised to the", "type": "text"}], "index": 32}], "index": 29.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [239, 178, 372, 209], "lines": [{"bbox": [239, 178, 372, 209], "spans": [{"bbox": [239, 178, 372, 209], "score": 0.95, "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}\|a\\!-\\!b\|\\leq c\\leq\\operatorname{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 114], "lines": [{"bbox": [71, 73, 540, 89], "spans": [{"bbox": [71, 75, 182, 88], "score": 0.92, "content": "\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c", "type": "inline_equation", "height": 13, "width": 111}, {"bbox": [182, 73, 383, 89], "score": 1.0, "content": " is an unimportant sign independent of ", "type": "text"}, {"bbox": [383, 75, 391, 84], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [391, 73, 540, 89], "score": 1.0, "content": ". This Galois action will play", "type": "text"}], "index": 0}, {"bbox": [70, 88, 542, 103], "spans": [{"bbox": [70, 88, 321, 103], "score": 1.0, "content": "a fairly important role in this paper. Note that ", "type": "text"}, {"bbox": [321, 90, 367, 101], "score": 0.93, "content": "\\sigma_{-1}=C", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [367, 88, 542, 103], "score": 1.0, "content": ", so this action can be thought of", "type": "text"}], "index": 1}, {"bbox": [70, 101, 458, 118], "spans": [{"bbox": [70, 101, 370, 118], "score": 1.0, "content": "as a generalisation of charge-conjugation. Note also that ", "type": "text"}, {"bbox": [371, 102, 453, 115], "score": 0.93, "content": "\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [453, 101, 458, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 73, 542, 118]}, {"type": "text", "bbox": [69, 115, 541, 164], "lines": [{"bbox": [94, 115, 541, 132], "spans": [{"bbox": [94, 115, 541, 132], "score": 1.0, "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23", "type": "text"}], "index": 3}, {"bbox": [69, 132, 541, 146], "spans": [{"bbox": [69, 132, 541, 146], "score": 1.0, "content": "p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities", "type": "text"}], "index": 4}, {"bbox": [71, 146, 255, 168], "spans": [{"bbox": [71, 146, 215, 166], "score": 0.93, "content": "T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))", "type": "inline_equation", "height": 20, "width": 144}, {"bbox": [215, 148, 232, 168], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [232, 151, 248, 162], "score": 0.9, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 148, 255, 168], "score": 1.0, "content": ":", "type": "text"}], "index": 5}], "index": 4, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [69, 115, 541, 168]}, {"type": "interline_equation", "bbox": [239, 178, 372, 209], "lines": [{"bbox": [239, 178, 372, 209], "spans": [{"bbox": [239, 178, 372, 209], "score": 0.95, "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "type": "interline_equation"}], "index": 6}], "index": 6, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 221, 541, 281], "lines": [{"bbox": [66, 218, 539, 247], "spans": [{"bbox": [66, 218, 105, 247], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 223, 206, 240], "score": 0.92, "content": "w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho", "type": "inline_equation", "height": 17, "width": 101}, {"bbox": [207, 218, 231, 247], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 227, 245, 237], "score": 0.81, "content": "W", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [245, 218, 388, 247], "score": 1.0, "content": " is the affine Weyl group of ", "type": "text"}, {"bbox": [388, 223, 412, 238], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [412, 218, 517, 247], "score": 1.0, "content": "(the dependence of ", "type": "text"}, {"bbox": [518, 228, 539, 242], "score": 0.92, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 21}], "index": 7}, {"bbox": [69, 239, 541, 256], "spans": [{"bbox": [69, 239, 88, 256], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [88, 242, 95, 251], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [96, 239, 249, 256], "score": 1.0, "content": " arises through the action of ", "type": "text"}, {"bbox": [249, 243, 262, 252], "score": 0.85, "content": "W", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [263, 239, 541, 256], "score": 1.0, "content": "). We shall see shortly that these fusion coefficients,", "type": "text"}], "index": 8}, {"bbox": [69, 254, 541, 270], "spans": [{"bbox": [69, 254, 358, 270], "score": 1.0, "content": "now manifestly integral, are in fact nonnegative. Let ", "type": "text"}, {"bbox": [359, 256, 401, 268], "score": 0.95, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [402, 254, 541, 270], "score": 1.0, "content": " denote the corresponding", "type": "text"}], "index": 9}, {"bbox": [70, 268, 131, 285], "spans": [{"bbox": [70, 268, 131, 285], "score": 1.0, "content": "fusion ring.", "type": "text"}], "index": 10}], "index": 8.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [66, 218, 541, 285]}, {"type": "text", "bbox": [70, 282, 541, 313], "lines": [{"bbox": [93, 283, 539, 301], "spans": [{"bbox": [93, 283, 330, 301], "score": 1.0, "content": "A handy consequence of (2.4) that whenever ", "type": "text"}, {"bbox": [330, 287, 337, 296], "score": 0.91, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [338, 283, 448, 301], "score": 1.0, "content": " is large enough that ", "type": "text"}, {"bbox": [448, 283, 539, 300], "score": 0.92, "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 91}], "index": 11}, {"bbox": [69, 297, 335, 318], "spans": [{"bbox": [69, 297, 120, 318], "score": 1.0, "content": "(i.e. that", "type": "text"}, {"bbox": [121, 300, 237, 314], "score": 0.91, "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "type": "inline_equation", "height": 14, "width": 116}, {"bbox": [237, 297, 272, 318], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [273, 302, 330, 315], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [330, 297, 335, 318], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [69, 283, 539, 318]}, {"type": "text", "bbox": [70, 313, 541, 354], "lines": [{"bbox": [93, 313, 540, 327], "spans": [{"bbox": [93, 313, 540, 327], "score": 1.0, "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "type": "text"}], "index": 13}, {"bbox": [69, 327, 541, 344], "spans": [{"bbox": [69, 327, 154, 343], "score": 1.0, "content": "fusion matrices ", "type": "text"}, {"bbox": [155, 330, 171, 340], "score": 0.91, "content": "N_{\\lambda}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [171, 327, 236, 343], "score": 1.0, "content": ", defined by ", "type": "text"}, {"bbox": [236, 329, 311, 344], "score": 0.9, "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 15, "width": 75}, {"bbox": [312, 327, 392, 343], "score": 1.0, "content": ". For instance, ", "type": "text"}, {"bbox": [392, 330, 430, 341], "score": 0.94, "content": "N_{0}=I", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [430, 327, 541, 343], "score": 1.0, "content": " and, more generally,", "type": "text"}], "index": 14}, {"bbox": [71, 343, 315, 357], "spans": [{"bbox": [71, 344, 86, 357], "score": 0.91, "content": "N_{j}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [86, 343, 301, 356], "score": 1.0, "content": " is the permutation matrix associated to ", "type": "text"}, {"bbox": [302, 344, 309, 353], "score": 0.82, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [310, 343, 315, 356], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [69, 313, 541, 357]}, {"type": "text", "bbox": [70, 355, 540, 383], "lines": [{"bbox": [93, 355, 542, 374], "spans": [{"bbox": [93, 355, 542, 374], "score": 1.0, "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "type": "text"}], "index": 16}, {"bbox": [70, 371, 112, 387], "spans": [{"bbox": [70, 371, 112, 387], "score": 1.0, "content": "fusions:", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 355, 542, 387]}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 463, 235, 477], "lines": [{"bbox": [70, 465, 233, 479], "spans": [{"bbox": [70, 465, 195, 479], "score": 1.0, "content": "for any simple-currents ", "type": "text"}, {"bbox": [195, 465, 230, 478], "score": 0.92, "content": "J,J^{\\prime},j", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [230, 465, 233, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 465, 233, 479]}, {"type": "text", "bbox": [70, 478, 541, 572], "lines": [{"bbox": [93, 479, 542, 495], "spans": [{"bbox": [93, 479, 190, 495], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [190, 480, 233, 494], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [234, 479, 312, 495], "score": 1.0, "content": " we may take ", "type": "text"}, {"bbox": [313, 479, 414, 493], "score": 0.89, "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [414, 479, 493, 495], "score": 1.0, "content": " (the value of ", "type": "text"}, {"bbox": [493, 482, 506, 492], "score": 0.85, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [506, 479, 542, 495], "score": 1.0, "content": "), and", "type": "text"}], "index": 20}, {"bbox": [64, 489, 548, 523], "spans": [{"bbox": [64, 489, 548, 523], "score": 1.0, "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(\u03c0 (a+1k)+ (2b+1) ). Charge-conjugation C is", "type": "text"}], "index": 21}, {"bbox": [70, 515, 542, 530], "spans": [{"bbox": [70, 515, 159, 530], "score": 1.0, "content": "trivial here, but ", "type": "text"}, {"bbox": [159, 516, 189, 529], "score": 0.9, "content": "j=k", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [190, 515, 456, 530], "score": 1.0, "content": " is a simple-current corresponding to permutation ", "type": "text"}, {"bbox": [456, 517, 516, 527], "score": 0.91, "content": "J a=k-a", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [516, 515, 542, 530], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [69, 527, 542, 547], "spans": [{"bbox": [69, 527, 117, 547], "score": 1.0, "content": "function ", "type": "text"}, {"bbox": [118, 531, 182, 544], "score": 0.92, "content": "Q_{j}(a)=a/2", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [182, 527, 317, 547], "score": 1.0, "content": ". The Galois action sends ", "type": "text"}, {"bbox": [317, 533, 325, 541], "score": 0.69, "content": "a", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [325, 527, 437, 547], "score": 1.0, "content": " to the unique weight ", "type": "text"}, {"bbox": [438, 529, 486, 543], "score": 0.92, "content": "a^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 48}, {"bbox": [487, 527, 542, 547], "score": 1.0, "content": " satisfying", "type": "text"}], "index": 23}, {"bbox": [71, 542, 542, 561], "spans": [{"bbox": [71, 544, 185, 558], "score": 0.91, "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [185, 542, 221, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [221, 544, 257, 557], "score": 0.46, "content": "2k+4", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [258, 542, 425, 561], "score": 1.0, "content": "), where that sign there equals ", "type": "text"}, {"bbox": [425, 545, 471, 558], "score": 0.93, "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [472, 542, 542, 561], "score": 1.0, "content": ". The fusion", "type": "text"}], "index": 24}, {"bbox": [71, 559, 196, 574], "spans": [{"bbox": [71, 559, 196, 574], "score": 1.0, "content": "coefficients are given by", "type": "text"}], "index": 25}], "index": 22.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [64, 479, 548, 574]}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}\|a\\!-\\!b\|\\leq c\\leq\\operatorname{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 624, 541, 716], "lines": [{"bbox": [95, 627, 541, 642], "spans": [{"bbox": [95, 627, 541, 642], "score": 1.0, "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "type": "text"}], "index": 27}, {"bbox": [71, 640, 542, 656], "spans": [{"bbox": [71, 642, 131, 655], "score": 0.91, "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 640, 168, 656], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [169, 642, 209, 655], "score": 0.9, "content": "\\operatorname{Ch}(X_{r})", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [209, 640, 450, 656], "score": 1.0, "content": " is the character ring for all finite-dimensional ", "type": "text"}, {"bbox": [451, 643, 466, 654], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [466, 640, 542, 656], "score": 1.0, "content": "-modules, and", "type": "text"}], "index": 28}, {"bbox": [71, 656, 541, 671], "spans": [{"bbox": [71, 658, 85, 668], "score": 0.91, "content": "\\mathcal{I}_{k}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [86, 656, 300, 671], "score": 1.0, "content": " is the subspace spanned by the elements ", "type": "text"}, {"bbox": [300, 657, 397, 670], "score": 0.92, "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [397, 656, 541, 671], "score": 1.0, "content": ". Finkelberg [8] proved that", "type": "text"}], "index": 29}, {"bbox": [70, 671, 540, 687], "spans": [{"bbox": [70, 672, 363, 687], "score": 1.0, "content": "this ring is isomorphic to the K-ring of a \u201csub-quotient\u201d", "type": "text"}, {"bbox": [363, 671, 379, 685], "score": 0.91, "content": "\\widetilde{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [379, 672, 540, 687], "score": 1.0, "content": " of Kazhdan-Lusztig\u2019s category", "type": "text"}], "index": 30}, {"bbox": [69, 684, 539, 704], "spans": [{"bbox": [69, 684, 111, 704], "score": 1.0, "content": "of level ", "type": "text"}, {"bbox": [112, 690, 119, 699], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [119, 684, 257, 704], "score": 1.0, "content": " integrable highest weight ", "type": "text"}, {"bbox": [257, 686, 281, 700], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [281, 684, 524, 704], "score": 1.0, "content": "-modules, and t o Gelfand-Kazhdan\u2019s category", "type": "text"}, {"bbox": [525, 687, 539, 702], "score": 0.92, "content": "\\widetilde{\\mathcal{O}}_{q}", "type": "inline_equation", "height": 15, "width": 14}], "index": 31}, {"bbox": [71, 703, 541, 717], "spans": [{"bbox": [71, 703, 412, 717], "score": 1.0, "content": "coming from finite-dimensional modules of the quantum group ", "type": "text"}, {"bbox": [413, 704, 441, 717], "score": 0.93, "content": "U_{q}X_{r}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [442, 703, 541, 717], "score": 1.0, "content": " specialised to the", "type": "text"}], "index": 32}, {"bbox": [70, 73, 541, 89], "spans": [{"bbox": [70, 73, 143, 89], "score": 1.0, "content": "root of unity ", "type": "text", "cross_page": true}, {"bbox": [144, 75, 192, 87], "score": 0.94, "content": "q\\,=\\,\\xi_{2m\\kappa}", "type": "inline_equation", "height": 12, "width": 48, "cross_page": true}, {"bbox": [192, 73, 330, 89], "score": 1.0, "content": " for appropriate choice of ", "type": "text", "cross_page": true}, {"bbox": [331, 75, 399, 87], "score": 0.94, "content": "m\\in\\{1,2,3\\}", "type": "inline_equation", "height": 12, "width": 68, "cross_page": true}, {"bbox": [399, 73, 541, 89], "score": 1.0, "content": ". They also arise from the", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 88, 541, 103], "spans": [{"bbox": [70, 88, 403, 103], "score": 1.0, "content": "Huang-Lepowsky coproduct [21] for the modules of the VOA ", "type": "text", "cross_page": true}, {"bbox": [404, 89, 439, 101], "score": 0.94, "content": "L(k,0)", "type": "inline_equation", "height": 12, "width": 35, "cross_page": true}, {"bbox": [440, 88, 541, 103], "score": 1.0, "content": ". Because of these", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [70, 103, 481, 118], "spans": [{"bbox": [70, 103, 233, 117], "score": 1.0, "content": "isomorphisms, we get that the ", "type": "text", "cross_page": true}, {"bbox": [233, 104, 255, 118], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 22, "cross_page": true}, {"bbox": [255, 103, 343, 117], "score": 1.0, "content": " do indeed lie in ", "type": "text", "cross_page": true}, {"bbox": [343, 104, 360, 117], "score": 0.9, "content": "\\mathbb{Z}_{\\geq}", "type": "inline_equation", "height": 13, "width": 17, "cross_page": true}, {"bbox": [360, 103, 481, 117], "score": 1.0, "content": ", for any affine algebra.", "type": "text", "cross_page": true}], "index": 2}], "index": 29.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [69, 627, 542, 717]}]}
0002044v1	6	For instance $$\pi$$ must send $$J$$ -fixed-points to $$\pi(J)$$ -fixed-points. More generally, a fusion-homomorphism $$\pi$$ is defined in the obvious algebraic way. It turns out that for such a $$\pi$$ , $$\pi\lambda\,=\,\pi\mu$$ iff $$\mu\,=\,J\lambda$$ for some simple-current $$J$$ for which $$\pi(J0)=\tilde{0}$$ . Moreover, $$\pi(J0)=\tilde{0}$$ is possible only if there are no $$J$$ -fixed-points. When $$\pi$$ is one-to-one (e.g. when there are no nontrivial simple-currents in $$P_{+}^{k}(X_{r}^{(1)}))$$ , then $$\pi$$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere. The key to finding fusion-symmetries is the following Lemma. Lemma 2.2. Let $$\widetilde{S}$$ be the Kac-Peterson matrix for $$Y_{s}^{(1)}$$ level $$m$$ . Then a bijection $$\pi\,:\,P_{+}^{k}(X_{r}^{(1)})\,\to\,P_{+}^{m}(Y_{s}^{(1)})$$ defines an isomorphism of fusion rings iff there exists some bijection $$\pi^{\prime}:P_{+}^{k}(X_{r}^{(1)})\to P_{+}^{m}(Y_{s}^{(1)})$$ such that $$S_{\lambda\mu}=\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}$$ for all $$\lambda,\mu\in P_{+}^{k}(X_{r}^{(1)})$$ . In particular, a permutation $$\pi$$ is a fusion-symmetry iff $$(\pi,\pi^{\prime})$$ is an $$S$$ -symmetry for some $$\pi^{\prime}$$ . Proof. The equality $$N_{\lambda\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}$$ means that, for each $$\mu$$ , the column vectors $$(\underline{{x}}_{\mu})_{\nu}=$$ $$\widetilde{S}_{\pi\nu,\pi\mu}$$ are simultaneous eigenvectors for the fusion matrices $$N_{\lambda}$$ , with eigenvalues $$\widetilde{S}_{\pi\lambda,\pi\mu}/\widetilde{S}_{0,\pi\mu}$$ . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $$S$$ . Thus there must be a permutation $$\pi^{\prime\prime}$$ of $$P_{+}^{k}(X_{r}^{(1)})$$ and scalars $$\alpha(\mu)$$ such that $$\widetilde{S}_{\pi\nu,\pi\mu}\,=\,\alpha(\mu)\,S_{\nu,\pi^{\prime\prime}\mu}$$ . Taking $$\nu=0$$ forces $$\alpha(\mu)>0$$ , and then unitarity forces $$\alpha(\mu)=1$$ . ■ Let $$\pi$$ be any isomorphism, and let $$\pi^{\prime}$$ be as in the Lemma. Then $$\pi^{\prime}$$ is also an isomorphism, with $$(\pi^{\prime})^{\prime}\;=\;\pi$$ . Equation (2.2b) implies for all $$\lambda\:\in\:P_{+}$$ and all simple- currents $$j$$ , that Another quick consequence of the Lemma is that for any Galois automorphism $$\sigma_{\ell}$$ and isomorphism $$\pi$$ , we have $$\tilde{\epsilon}_{\ell}(\pi\lambda)=\epsilon_{\ell}(\lambda)$$ and $$\pi(\lambda^{(\ell)})=(\pi\lambda)^{(\ell)}$$ . To see this, apply the invertibility of $$S$$ to the equation A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $$\Gamma=\{\gamma_{1},...,\gamma_{m}\}$$ of $$P_{+}$$ which generates $$\mathcal{R}(X_{r,k})$$ as a ring. Diagonalising, this is equivalent to requiring that there are $$m$$ -variable polynomials $$P_{\lambda}(x_{1},\ldots,x_{m})$$ such that Let $$(\pi,\pi^{\prime})$$ be an $$S$$ -symmetry, and suppose we know that $$\pi\gamma=\gamma$$ for all $$\gamma$$ in the fusion- generator $$\Gamma$$ . Then for any $$\lambda\in P_{+}$$ , for all $$\mu\in P_{+}$$ , so $$\pi\lambda=\lambda$$ . One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character	<p>For instance $$\pi$$ must send $$J$$ -fixed-points to $$\pi(J)$$ -fixed-points.</p> <p>More generally, a fusion-homomorphism $$\pi$$ is defined in the obvious algebraic way. It turns out that for such a $$\pi$$ , $$\pi\lambda\,=\,\pi\mu$$ iff $$\mu\,=\,J\lambda$$ for some simple-current $$J$$ for which $$\pi(J0)=\tilde{0}$$ . Moreover, $$\pi(J0)=\tilde{0}$$ is possible only if there are no $$J$$ -fixed-points. When $$\pi$$ is one-to-one (e.g. when there are no nontrivial simple-currents in $$P_{+}^{k}(X_{r}^{(1)}))$$ , then $$\pi$$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere.</p> <p>The key to finding fusion-symmetries is the following Lemma.</p> <p>Lemma 2.2. Let $$\widetilde{S}$$ be the Kac-Peterson matrix for $$Y_{s}^{(1)}$$ level $$m$$ . Then a bijection $$\pi\,:\,P_{+}^{k}(X_{r}^{(1)})\,\to\,P_{+}^{m}(Y_{s}^{(1)})$$ defines an isomorphism of fusion rings iff there exists some bijection $$\pi^{\prime}:P_{+}^{k}(X_{r}^{(1)})\to P_{+}^{m}(Y_{s}^{(1)})$$ such that $$S_{\lambda\mu}=\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}$$ for all $$\lambda,\mu\in P_{+}^{k}(X_{r}^{(1)})$$ . In particular, a permutation $$\pi$$ is a fusion-symmetry iff $$(\pi,\pi^{\prime})$$ is an $$S$$ -symmetry for some $$\pi^{\prime}$$ .</p> <p>Proof. The equality $$N_{\lambda\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}$$ means that, for each $$\mu$$ , the column vectors $$(\underline{{x}}_{\mu})_{\nu}=$$ $$\widetilde{S}_{\pi\nu,\pi\mu}$$ are simultaneous eigenvectors for the fusion matrices $$N_{\lambda}$$ , with eigenvalues $$\widetilde{S}_{\pi\lambda,\pi\mu}/\widetilde{S}_{0,\pi\mu}$$ . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $$S$$ . Thus there must be a permutation $$\pi^{\prime\prime}$$ of $$P_{+}^{k}(X_{r}^{(1)})$$ and scalars $$\alpha(\mu)$$ such that $$\widetilde{S}_{\pi\nu,\pi\mu}\,=\,\alpha(\mu)\,S_{\nu,\pi^{\prime\prime}\mu}$$ . Taking $$\nu=0$$ forces $$\alpha(\mu)>0$$ , and then unitarity forces $$\alpha(\mu)=1$$ . ■</p> <p>Let $$\pi$$ be any isomorphism, and let $$\pi^{\prime}$$ be as in the Lemma. Then $$\pi^{\prime}$$ is also an isomorphism, with $$(\pi^{\prime})^{\prime}\;=\;\pi$$ . Equation (2.2b) implies for all $$\lambda\:\in\:P_{+}$$ and all simple- currents $$j$$ , that</p> <p>Another quick consequence of the Lemma is that for any Galois automorphism $$\sigma_{\ell}$$ and isomorphism $$\pi$$ , we have $$\tilde{\epsilon}_{\ell}(\pi\lambda)=\epsilon_{\ell}(\lambda)$$ and $$\pi(\lambda^{(\ell)})=(\pi\lambda)^{(\ell)}$$ . To see this, apply the invertibility of $$S$$ to the equation</p> <p>A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $$\Gamma=\{\gamma_{1},...,\gamma_{m}\}$$ of $$P_{+}$$ which generates $$\mathcal{R}(X_{r,k})$$ as a ring. Diagonalising, this is equivalent to requiring that there are $$m$$ -variable polynomials $$P_{\lambda}(x_{1},\ldots,x_{m})$$ such that</p> <p>Let $$(\pi,\pi^{\prime})$$ be an $$S$$ -symmetry, and suppose we know that $$\pi\gamma=\gamma$$ for all $$\gamma$$ in the fusion- generator $$\Gamma$$ . Then for any $$\lambda\in P_{+}$$ ,</p> <p>for all $$\mu\in P_{+}$$ , so $$\pi\lambda=\lambda$$ .</p> <p>One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character</p>	[{"type": "text", "coordinates": [70, 70, 392, 85], "content": "For instance $$\\pi$$ must send $$J$$ -fixed-points to $$\\pi(J)$$ -fixed-points.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 86, 542, 160], "content": "More generally, a fusion-homomorphism $$\\pi$$ is defined in the obvious algebraic way. It\nturns out that for such a $$\\pi$$ , $$\\pi\\lambda\\,=\\,\\pi\\mu$$ iff $$\\mu\\,=\\,J\\lambda$$ for some simple-current $$J$$ for which\n$$\\pi(J0)=\\tilde{0}$$ . Moreover, $$\\pi(J0)=\\tilde{0}$$ is possible only if there are no $$J$$ -fixed-points. When $$\\pi$$ is\none-to-one (e.g. when there are no nontrivial simple-currents in $$P_{+}^{k}(X_{r}^{(1)}))$$ , then $$\\pi$$ obeys\n(2.6). Fusion-homomorphisms will be studied elsewhere.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [95, 160, 420, 174], "content": "The key to finding fusion-symmetries is the following Lemma.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 180, 543, 246], "content": "Lemma 2.2. Let $$\\widetilde{S}$$ be the Kac-Peterson matrix for $$Y_{s}^{(1)}$$ level $$m$$ . Then a bijection\n$$\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})$$ defines an isomorphism of fusion rings iff there exists some\nbijection $$\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})$$ such that $$S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}$$ for all $$\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})$$ . In\nparticular, a permutation $$\\pi$$ is a fusion-symmetry iff $$(\\pi,\\pi^{\\prime})$$ is an $$S$$ -symmetry for some $$\\pi^{\\prime}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 250, 558, 344], "content": "Proof. The equality $$N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}$$ means that, for each $$\\mu$$ , the column vectors $$(\\underline{{x}}_{\\mu})_{\\nu}=$$\n$$\\widetilde{S}_{\\pi\\nu,\\pi\\mu}$$ are simultaneous eigenvectors for the fusion matrices $$N_{\\lambda}$$ , with eigenvalues $$\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}$$ .\nI t is easy to see from Verlinde\u2019s formula (1.1b) that any simultaneous eigenvec t or for a ll\nfusion matrices must be a scalar multiple of some column of $$S$$ . Thus there must be a\npermutation $$\\pi^{\\prime\\prime}$$ of $$P_{+}^{k}(X_{r}^{(1)})$$ and scalars $$\\alpha(\\mu)$$ such that $$\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}$$ . Taking\n$$\\nu=0$$ forces $$\\alpha(\\mu)>0$$ , and then unitarity forces $$\\alpha(\\mu)=1$$ . \u25a0", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [70, 349, 541, 391], "content": "Let $$\\pi$$ be any isomorphism, and let $$\\pi^{\\prime}$$ be as in the Lemma. Then $$\\pi^{\\prime}$$ is also an\nisomorphism, with $$(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi$$ . Equation (2.2b) implies for all $$\\lambda\\:\\in\\:P_{+}$$ and all simple-\ncurrents $$j$$ , that", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [190, 391, 422, 409], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 414, 541, 458], "content": "Another quick consequence of the Lemma is that for any Galois automorphism $$\\sigma_{\\ell}$$\nand isomorphism $$\\pi$$ , we have $$\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)$$ and $$\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}$$ . To see this, apply the\ninvertibility of $$S$$ to the equation", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [111, 468, 507, 488], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [69, 496, 542, 540], "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a\nsubset $$\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}$$ of $$P_{+}$$ which generates $$\\mathcal{R}(X_{r,k})$$ as a ring. Diagonalising, this is\nequivalent to requiring that there are $$m$$ -variable polynomials $$P_{\\lambda}(x_{1},\\ldots,x_{m})$$ such that", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [189, 552, 423, 582], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [69, 592, 541, 621], "content": "Let $$(\\pi,\\pi^{\\prime})$$ be an $$S$$ -symmetry, and suppose we know that $$\\pi\\gamma=\\gamma$$ for all $$\\gamma$$ in the fusion-\ngenerator $$\\Gamma$$ . Then for any $$\\lambda\\in P_{+}$$ ,", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [146, 632, 465, 663], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [69, 672, 207, 686], "content": "for all $$\\mu\\in P_{+}$$ , so $$\\pi\\lambda=\\lambda$$ .", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [70, 687, 541, 715], "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the\nexistence of small fusion-generators. In particular, because we know that any Lie character", "block_type": "text", "index": 15}]	[{"type": "text", "coordinates": [70, 74, 139, 87], "content": "For instance ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [139, 79, 146, 84], "content": "\\pi", "score": 0.86, "index": 2}, {"type": "text", "coordinates": [147, 74, 207, 87], "content": " must send ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [208, 75, 216, 84], "content": "J", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [216, 74, 298, 87], "content": "-fixed-points to ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [298, 75, 323, 87], "content": "\\pi(J)", "score": 0.94, "index": 6}, {"type": "text", "coordinates": [324, 74, 390, 87], "content": "-fixed-points.", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [95, 88, 309, 102], "content": "More generally, a fusion-homomorphism ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [309, 93, 316, 99], "content": "\\pi", "score": 0.85, "index": 9}, {"type": "text", "coordinates": [317, 88, 541, 102], "content": " is defined in the obvious algebraic way. It", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [70, 102, 212, 117], "content": "turns out that for such a ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [213, 108, 220, 113], "content": "\\pi", "score": 0.82, "index": 12}, {"type": "text", "coordinates": [220, 102, 228, 117], "content": ", ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [228, 104, 278, 115], "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [278, 102, 298, 117], "content": " iff", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [298, 104, 341, 115], "content": "\\mu\\,=\\,J\\lambda", "score": 0.93, "index": 16}, {"type": "text", "coordinates": [341, 102, 476, 117], "content": " for some simple-current ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [477, 105, 484, 113], "content": "J", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [485, 102, 541, 117], "content": " for which", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [71, 117, 124, 131], "content": "\\pi(J0)=\\tilde{0}", "score": 0.94, "index": 20}, {"type": "text", "coordinates": [124, 116, 187, 132], "content": ". Moreover, ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [187, 117, 240, 131], "content": "\\pi(J0)=\\tilde{0}", "score": 0.95, "index": 22}, {"type": "text", "coordinates": [241, 116, 404, 132], "content": " is possible only if there are no ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [404, 119, 412, 127], "content": "J", "score": 0.9, "index": 24}, {"type": "text", "coordinates": [413, 116, 520, 132], "content": "-fixed-points. When ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [520, 122, 528, 127], "content": "\\pi", "score": 0.88, "index": 26}, {"type": "text", "coordinates": [528, 116, 542, 132], "content": " is", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [69, 131, 409, 149], "content": "one-to-one (e.g. when there are no nontrivial simple-currents in ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [410, 131, 463, 148], "content": "P_{+}^{k}(X_{r}^{(1)}))", "score": 0.93, "index": 29}, {"type": "text", "coordinates": [463, 131, 498, 149], "content": ", then ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [498, 138, 506, 144], "content": "\\pi", "score": 0.88, "index": 31}, {"type": "text", "coordinates": [506, 131, 542, 149], "content": " obeys", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [72, 148, 366, 162], "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [95, 162, 419, 176], "content": "The key to finding fusion-symmetries is the following Lemma.", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [91, 178, 194, 201], "content": "Lemma 2.2. Let", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [194, 183, 203, 196], "content": "\\widetilde{S}", "score": 0.87, "index": 36}, {"type": "text", "coordinates": [203, 178, 376, 201], "content": " be the Kac-Peterson matrix for ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [376, 181, 400, 197], "content": "Y_{s}^{(1)}", "score": 0.9, "index": 38}, {"type": "text", "coordinates": [400, 178, 431, 201], "content": "level ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [431, 186, 443, 196], "content": "m", "score": 0.46, "index": 40}, {"type": "text", "coordinates": [443, 178, 543, 201], "content": ". Then a bijection", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [71, 198, 214, 216], "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [214, 195, 543, 218], "content": " defines an isomorphism of fusion rings iff there exists some", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [68, 212, 119, 236], "content": "bijection ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [120, 216, 261, 233], "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "score": 0.92, "index": 45}, {"type": "text", "coordinates": [261, 212, 317, 236], "content": " such that ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [317, 216, 392, 232], "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "score": 0.94, "index": 47}, {"type": "text", "coordinates": [392, 212, 433, 236], "content": " for all ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [433, 215, 518, 233], "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "score": 0.91, "index": 49}, {"type": "text", "coordinates": [518, 212, 544, 236], "content": ". In", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [70, 232, 205, 247], "content": "particular, a permutation ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [206, 238, 213, 243], "content": "\\pi", "score": 0.67, "index": 52}, {"type": "text", "coordinates": [213, 232, 344, 247], "content": " is a fusion-symmetry iff", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [344, 233, 377, 246], "content": "(\\pi,\\pi^{\\prime})", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [377, 232, 409, 247], "content": " is an ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [409, 234, 418, 244], "content": "S", "score": 0.83, "index": 56}, {"type": "text", "coordinates": [418, 232, 524, 247], "content": "-symmetry for some ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [525, 234, 536, 243], "content": "\\pi^{\\prime}", "score": 0.87, "index": 58}, {"type": "text", "coordinates": [536, 232, 541, 247], "content": ".", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [68, 250, 185, 273], "content": "Proof. The equality ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [185, 256, 263, 270], "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "score": 0.93, "index": 61}, {"type": "text", "coordinates": [263, 250, 378, 273], "content": " means that, for each ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [378, 256, 387, 267], "content": "\\mu", "score": 0.75, "index": 63}, {"type": "text", "coordinates": [387, 250, 497, 273], "content": ", the column vectors ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [498, 254, 542, 269], "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "score": 0.9, "index": 65}, {"type": "inline_equation", "coordinates": [71, 270, 105, 286], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "score": 0.92, "index": 66}, {"type": "text", "coordinates": [105, 268, 377, 290], "content": " are simultaneous eigenvectors for the fusion matrices", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [378, 272, 394, 284], "content": "N_{\\lambda}", "score": 0.89, "index": 68}, {"type": "text", "coordinates": [394, 268, 486, 290], "content": ", with eigenvalues", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [487, 269, 555, 286], "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [555, 268, 559, 290], "content": ".", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [70, 285, 542, 301], "content": "I t is easy to see from Verlinde\u2019s formula (1.1b) that any simultaneous eigenvec t or for a ll", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [71, 300, 402, 315], "content": "fusion matrices must be a scalar multiple of some column of ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [402, 302, 410, 311], "content": "S", "score": 0.85, "index": 74}, {"type": "text", "coordinates": [411, 300, 542, 315], "content": ". Thus there must be a", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [68, 312, 140, 335], "content": "permutation ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [141, 317, 155, 327], "content": "\\pi^{\\prime\\prime}", "score": 0.9, "index": 77}, {"type": "text", "coordinates": [155, 312, 174, 335], "content": " of ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [174, 314, 223, 331], "content": "P_{+}^{k}(X_{r}^{(1)})", "score": 0.95, "index": 79}, {"type": "text", "coordinates": [224, 312, 293, 335], "content": " and scalars ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [293, 317, 317, 330], "content": "\\alpha(\\mu)", "score": 0.94, "index": 81}, {"type": "text", "coordinates": [318, 312, 378, 335], "content": " such that", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [378, 315, 491, 330], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "score": 0.92, "index": 83}, {"type": "text", "coordinates": [492, 312, 543, 335], "content": ". Taking", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [71, 333, 100, 342], "content": "\\nu=0", "score": 0.91, "index": 85}, {"type": "text", "coordinates": [100, 331, 137, 345], "content": " forces ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [137, 332, 184, 344], "content": "\\alpha(\\mu)>0", "score": 0.94, "index": 87}, {"type": "text", "coordinates": [184, 331, 325, 345], "content": ", and then unitarity forces ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [325, 332, 371, 344], "content": "\\alpha(\\mu)=1", "score": 0.94, "index": 89}, {"type": "text", "coordinates": [372, 331, 377, 345], "content": ". ", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [401, 332, 412, 344], "content": "\u25a0", "score": 0.9251790046691895, "index": 91}, {"type": "text", "coordinates": [92, 351, 118, 366], "content": "Let ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [119, 357, 126, 362], "content": "\\pi", "score": 0.86, "index": 93}, {"type": "text", "coordinates": [126, 351, 294, 366], "content": " be any isomorphism, and let ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [294, 353, 305, 362], "content": "\\pi^{\\prime}", "score": 0.9, "index": 95}, {"type": "text", "coordinates": [306, 351, 470, 366], "content": " be as in the Lemma. Then ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [470, 353, 481, 362], "content": "\\pi^{\\prime}", "score": 0.89, "index": 97}, {"type": "text", "coordinates": [482, 351, 541, 366], "content": " is also an", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [71, 367, 175, 381], "content": "isomorphism, with ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [175, 367, 227, 380], "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "score": 0.94, "index": 100}, {"type": "text", "coordinates": [228, 367, 410, 381], "content": ". Equation (2.2b) implies for all ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [411, 367, 453, 380], "content": "\\lambda\\:\\in\\:P_{+}", "score": 0.91, "index": 102}, {"type": "text", "coordinates": [454, 367, 538, 381], "content": " and all simple-", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [71, 382, 117, 393], "content": "currents ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [117, 383, 123, 393], "content": "j", "score": 0.88, "index": 105}, {"type": "text", "coordinates": [124, 382, 154, 393], "content": ", that", "score": 1.0, "index": 106}, {"type": "interline_equation", "coordinates": [190, 391, 422, 409], "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [94, 416, 528, 432], "content": "Another quick consequence of the Lemma is that for any Galois automorphism ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [528, 422, 539, 429], "content": "\\sigma_{\\ell}", "score": 0.86, "index": 109}, {"type": "text", "coordinates": [69, 430, 164, 447], "content": "and isomorphism ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [165, 436, 172, 442], "content": "\\pi", "score": 0.86, "index": 111}, {"type": "text", "coordinates": [173, 430, 225, 447], "content": ", we have ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [225, 432, 300, 444], "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "score": 0.94, "index": 113}, {"type": "text", "coordinates": [301, 430, 327, 447], "content": " and ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [328, 430, 416, 445], "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [416, 430, 542, 447], "content": ". To see this, apply the", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [71, 445, 149, 460], "content": "invertibility of ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [150, 447, 158, 456], "content": "S", "score": 0.91, "index": 118}, {"type": "text", "coordinates": [158, 445, 243, 460], "content": " to the equation", "score": 1.0, "index": 119}, {"type": "interline_equation", "coordinates": [111, 468, 507, 488], "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "score": 0.9, "index": 120}, {"type": "text", "coordinates": [95, 498, 542, 514], "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [70, 511, 108, 529], "content": "subset ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [108, 514, 199, 527], "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "score": 0.93, "index": 123}, {"type": "text", "coordinates": [200, 511, 217, 529], "content": " of ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [218, 515, 233, 526], "content": "P_{+}", "score": 0.92, "index": 125}, {"type": "text", "coordinates": [234, 511, 325, 529], "content": " which generates ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [325, 513, 369, 527], "content": "\\mathcal{R}(X_{r,k})", "score": 0.93, "index": 127}, {"type": "text", "coordinates": [369, 511, 542, 529], "content": " as a ring. Diagonalising, this is", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [72, 527, 270, 543], "content": "equivalent to requiring that there are ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [270, 532, 281, 538], "content": "m", "score": 0.78, "index": 130}, {"type": "text", "coordinates": [281, 527, 396, 543], "content": "-variable polynomials ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [396, 528, 474, 541], "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "score": 0.93, "index": 132}, {"type": "text", "coordinates": [474, 527, 529, 543], "content": " such that", "score": 1.0, "index": 133}, {"type": "interline_equation", "coordinates": [189, 552, 423, 582], "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "score": 0.93, "index": 134}, {"type": "text", "coordinates": [70, 594, 93, 609], "content": "Let ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [93, 595, 126, 608], "content": "(\\pi,\\pi^{\\prime})", "score": 0.94, "index": 136}, {"type": "text", "coordinates": [126, 594, 163, 609], "content": " be an ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [164, 596, 172, 605], "content": "S", "score": 0.9, "index": 138}, {"type": "text", "coordinates": [172, 594, 378, 609], "content": "-symmetry, and suppose we know that ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [379, 596, 418, 607], "content": "\\pi\\gamma=\\gamma", "score": 0.84, "index": 140}, {"type": "text", "coordinates": [419, 594, 457, 609], "content": " for all ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [457, 599, 465, 607], "content": "\\gamma", "score": 0.89, "index": 142}, {"type": "text", "coordinates": [465, 594, 540, 609], "content": " in the fusion-", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [70, 608, 124, 624], "content": "generator ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [124, 610, 132, 619], "content": "\\Gamma", "score": 0.88, "index": 145}, {"type": "text", "coordinates": [132, 608, 212, 624], "content": ". Then for any ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [213, 610, 250, 622], "content": "\\lambda\\in P_{+}", "score": 0.93, "index": 147}, {"type": "text", "coordinates": [250, 608, 254, 624], "content": ",", "score": 1.0, "index": 148}, {"type": "interline_equation", "coordinates": [146, 632, 465, 663], "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "score": 0.93, "index": 149}, {"type": "text", "coordinates": [71, 674, 106, 687], "content": "for all ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [106, 676, 144, 687], "content": "\\mu\\in P_{+}", "score": 0.93, "index": 151}, {"type": "text", "coordinates": [144, 674, 165, 687], "content": ", so ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [165, 676, 203, 685], "content": "\\pi\\lambda=\\lambda", "score": 0.93, "index": 153}, {"type": "text", "coordinates": [203, 674, 207, 687], "content": ".", "score": 1.0, "index": 154}, {"type": "text", "coordinates": [95, 689, 541, 703], "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "score": 1.0, "index": 155}, {"type": "text", "coordinates": [72, 703, 540, 717], "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "score": 1.0, "index": 156}]	[]	[{"type": "block", "coordinates": [190, 391, 422, 409], "content": "", "caption": ""}, {"type": "block", "coordinates": [111, 468, 507, 488], "content": "", "caption": ""}, {"type": "block", "coordinates": [189, 552, 423, 582], "content": "", "caption": ""}, {"type": "block", "coordinates": [146, 632, 465, 663], "content": "", "caption": ""}, {"type": "inline", "coordinates": [139, 79, 146, 84], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [208, 75, 216, 84], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [298, 75, 323, 87], "content": "\\pi(J)", "caption": ""}, {"type": "inline", "coordinates": [309, 93, 316, 99], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [213, 108, 220, 113], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [228, 104, 278, 115], "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "caption": ""}, {"type": "inline", "coordinates": [298, 104, 341, 115], "content": "\\mu\\,=\\,J\\lambda", "caption": ""}, {"type": "inline", "coordinates": [477, 105, 484, 113], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [71, 117, 124, 131], "content": "\\pi(J0)=\\tilde{0}", "caption": ""}, {"type": "inline", "coordinates": [187, 117, 240, 131], "content": "\\pi(J0)=\\tilde{0}", "caption": ""}, {"type": "inline", "coordinates": [404, 119, 412, 127], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [520, 122, 528, 127], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [410, 131, 463, 148], "content": "P_{+}^{k}(X_{r}^{(1)}))", "caption": ""}, {"type": "inline", "coordinates": [498, 138, 506, 144], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [194, 183, 203, 196], "content": "\\widetilde{S}", "caption": ""}, {"type": "inline", "coordinates": [376, 181, 400, 197], "content": "Y_{s}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [431, 186, 443, 196], "content": "m", "caption": ""}, {"type": "inline", "coordinates": [71, 198, 214, 216], "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [120, 216, 261, 233], "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [317, 216, 392, 232], "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "caption": ""}, {"type": "inline", "coordinates": [433, 215, 518, 233], "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [206, 238, 213, 243], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [344, 233, 377, 246], "content": "(\\pi,\\pi^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [409, 234, 418, 244], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [525, 234, 536, 243], "content": "\\pi^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [185, 256, 263, 270], "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "caption": ""}, {"type": "inline", "coordinates": [378, 256, 387, 267], "content": "\\mu", "caption": ""}, {"type": "inline", "coordinates": [498, 254, 542, 269], "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "caption": ""}, {"type": "inline", "coordinates": [71, 270, 105, 286], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "caption": ""}, {"type": "inline", "coordinates": [378, 272, 394, 284], "content": "N_{\\lambda}", "caption": ""}, {"type": "inline", "coordinates": [487, 269, 555, 286], "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "caption": ""}, {"type": "inline", "coordinates": [402, 302, 410, 311], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [141, 317, 155, 327], "content": "\\pi^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [174, 314, 223, 331], "content": "P_{+}^{k}(X_{r}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [293, 317, 317, 330], "content": "\\alpha(\\mu)", "caption": ""}, {"type": "inline", "coordinates": [378, 315, 491, 330], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "caption": ""}, {"type": "inline", "coordinates": [71, 333, 100, 342], "content": "\\nu=0", "caption": ""}, {"type": "inline", "coordinates": [137, 332, 184, 344], "content": "\\alpha(\\mu)>0", "caption": ""}, {"type": "inline", "coordinates": [325, 332, 371, 344], "content": "\\alpha(\\mu)=1", "caption": ""}, {"type": "inline", "coordinates": [119, 357, 126, 362], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [294, 353, 305, 362], "content": "\\pi^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [470, 353, 481, 362], "content": "\\pi^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [175, 367, 227, 380], "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "caption": ""}, {"type": "inline", "coordinates": [411, 367, 453, 380], "content": "\\lambda\\:\\in\\:P_{+}", "caption": ""}, {"type": "inline", "coordinates": [117, 383, 123, 393], "content": "j", "caption": ""}, {"type": "inline", "coordinates": [528, 422, 539, 429], "content": "\\sigma_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [165, 436, 172, 442], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [225, 432, 300, 444], "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "caption": ""}, {"type": "inline", "coordinates": [328, 430, 416, 445], "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "caption": ""}, {"type": "inline", "coordinates": [150, 447, 158, 456], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [108, 514, 199, 527], "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "caption": ""}, {"type": "inline", "coordinates": [218, 515, 233, 526], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [325, 513, 369, 527], "content": "\\mathcal{R}(X_{r,k})", "caption": ""}, {"type": "inline", "coordinates": [270, 532, 281, 538], "content": "m", "caption": ""}, {"type": "inline", "coordinates": [396, 528, 474, 541], "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "caption": ""}, {"type": "inline", "coordinates": [93, 595, 126, 608], "content": "(\\pi,\\pi^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [164, 596, 172, 605], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [379, 596, 418, 607], "content": "\\pi\\gamma=\\gamma", "caption": ""}, {"type": "inline", "coordinates": [457, 599, 465, 607], "content": "\\gamma", "caption": ""}, {"type": "inline", "coordinates": [124, 610, 132, 619], "content": "\\Gamma", "caption": ""}, {"type": "inline", "coordinates": [213, 610, 250, 622], "content": "\\lambda\\in P_{+}", "caption": ""}, {"type": "inline", "coordinates": [106, 676, 144, 687], "content": "\\mu\\in P_{+}", "caption": ""}, {"type": "inline", "coordinates": [165, 676, 203, 685], "content": "\\pi\\lambda=\\lambda", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "For instance $\\pi$ must send $J$ -fixed-points to $\\pi(J)$ -fixed-points. ", "page_idx": 6}, {"type": "text", "text": "More generally, a fusion-homomorphism $\\pi$ is defined in the obvious algebraic way. It turns out that for such a $\\pi$ , $\\pi\\lambda\\,=\\,\\pi\\mu$ iff $\\mu\\,=\\,J\\lambda$ for some simple-current $J$ for which $\\pi(J0)=\\tilde{0}$ . Moreover, $\\pi(J0)=\\tilde{0}$ is possible only if there are no $J$ -fixed-points. When $\\pi$ is one-to-one (e.g. when there are no nontrivial simple-currents in $P_{+}^{k}(X_{r}^{(1)}))$ , then $\\pi$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere. ", "page_idx": 6}, {"type": "text", "text": "The key to finding fusion-symmetries is the following Lemma. ", "page_idx": 6}, {"type": "text", "text": "Lemma 2.2. Let $\\widetilde{S}$ be the Kac-Peterson matrix for $Y_{s}^{(1)}$ level $m$ . Then a bijection $\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})$ defines an isomorphism of fusion rings iff there exists some bijection $\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})$ such that $S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}$ for all $\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})$ . In particular, a permutation $\\pi$ is a fusion-symmetry iff $(\\pi,\\pi^{\\prime})$ is an $S$ -symmetry for some $\\pi^{\\prime}$ . ", "page_idx": 6}, {"type": "text", "text": "Proof. The equality $N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}$ means that, for each $\\mu$ , the column vectors $(\\underline{{x}}_{\\mu})_{\\nu}=$ $\\widetilde{S}_{\\pi\\nu,\\pi\\mu}$ are simultaneous eigenvectors for the fusion matrices $N_{\\lambda}$ , with eigenvalues $\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}$ . I t is easy to see from Verlinde\u2019s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $S$ . Thus there must be a permutation $\\pi^{\\prime\\prime}$ of $P_{+}^{k}(X_{r}^{(1)})$ and scalars $\\alpha(\\mu)$ such that $\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}$ . Taking $\\nu=0$ forces $\\alpha(\\mu)>0$ , and then unitarity forces $\\alpha(\\mu)=1$ . \u25a0 ", "page_idx": 6}, {"type": "text", "text": "Let $\\pi$ be any isomorphism, and let $\\pi^{\\prime}$ be as in the Lemma. Then $\\pi^{\\prime}$ is also an isomorphism, with $(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi$ . Equation (2.2b) implies for all $\\lambda\\:\\in\\:P_{+}$ and all simplecurrents $j$ , that ", "page_idx": 6}, {"type": "equation", "text": "$$\nQ_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "Another quick consequence of the Lemma is that for any Galois automorphism $\\sigma_{\\ell}$ and isomorphism $\\pi$ , we have $\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)$ and $\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}$ . To see this, apply the invertibility of $S$ to the equation ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}$ of $P_{+}$ which generates $\\mathcal{R}(X_{r,k})$ as a ring. Diagonalising, this is equivalent to requiring that there are $m$ -variable polynomials $P_{\\lambda}(x_{1},\\ldots,x_{m})$ such that ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "Let $(\\pi,\\pi^{\\prime})$ be an $S$ -symmetry, and suppose we know that $\\pi\\gamma=\\gamma$ for all $\\gamma$ in the fusiongenerator $\\Gamma$ . Then for any $\\lambda\\in P_{+}$ , ", "page_idx": 6}, {"type": "equation", "text": "$$\n{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "for all $\\mu\\in P_{+}$ , so $\\pi\\lambda=\\lambda$ . ", "page_idx": 6}, {"type": "text", "text": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character $\\mathrm{ch}_{\\overline{{{\\mu}}}}$ for $X_{r}$ can be written as a polynomial in the fundamental characters $\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}$ , we know from (2.1b) that $\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}$ is a fusion-generator for $X_{r}^{(1)}$ at any level $k$ sufficiently large that $P_{+}$ contains all $\\Lambda_{i}$ (in other words, for any $k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $X_{r,k}$ is $\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}$ . Smaller fusion-generators usually exist \u2014 for example $\\{\\Lambda_{1}\\}$ is a fusion-generator for $A_{8,k}$ whenever $k$ is even and coprime to 3. 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It", "type": "text"}], "index": 1}, {"bbox": [70, 102, 541, 117], "spans": [{"bbox": [70, 102, 212, 117], "score": 1.0, "content": "turns out that for such a ", "type": "text"}, {"bbox": [213, 108, 220, 113], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [220, 102, 228, 117], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [228, 104, 278, 115], "score": 0.91, "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [278, 102, 298, 117], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [298, 104, 341, 115], "score": 0.93, "content": "\\mu\\,=\\,J\\lambda", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [341, 102, 476, 117], "score": 1.0, "content": " for some simple-current ", "type": "text"}, {"bbox": [477, 105, 484, 113], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [485, 102, 541, 117], "score": 1.0, "content": " for which", "type": "text"}], "index": 2}, {"bbox": [71, 116, 542, 132], "spans": [{"bbox": [71, 117, 124, 131], "score": 0.94, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [124, 116, 187, 132], "score": 1.0, "content": ". 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When ", "type": "text"}, {"bbox": [520, 122, 528, 127], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [528, 116, 542, 132], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [69, 131, 542, 149], "spans": [{"bbox": [69, 131, 409, 149], "score": 1.0, "content": "one-to-one (e.g. when there are no nontrivial simple-currents in ", "type": "text"}, {"bbox": [410, 131, 463, 148], "score": 0.93, "content": "P_{+}^{k}(X_{r}^{(1)}))", "type": "inline_equation", "height": 17, "width": 53}, {"bbox": [463, 131, 498, 149], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [498, 138, 506, 144], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [506, 131, 542, 149], "score": 1.0, "content": " obeys", "type": "text"}], "index": 4}, {"bbox": [72, 148, 366, 162], "spans": [{"bbox": [72, 148, 366, 162], "score": 1.0, "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "type": "text"}], "index": 5}], "index": 3}, {"type": "text", "bbox": [95, 160, 420, 174], "lines": [{"bbox": [95, 162, 419, 176], "spans": [{"bbox": [95, 162, 419, 176], "score": 1.0, "content": "The key to finding fusion-symmetries is the following Lemma.", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 180, 543, 246], "lines": [{"bbox": [91, 178, 543, 201], "spans": [{"bbox": [91, 178, 194, 201], "score": 1.0, "content": "Lemma 2.2. Let", "type": "text"}, {"bbox": [194, 183, 203, 196], "score": 0.87, "content": "\\widetilde{S}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [203, 178, 376, 201], "score": 1.0, "content": " be the Kac-Peterson matrix for ", "type": "text"}, {"bbox": [376, 181, 400, 197], "score": 0.9, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 16, "width": 24}, {"bbox": [400, 178, 431, 201], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [431, 186, 443, 196], "score": 0.46, "content": "m", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [443, 178, 543, 201], "score": 1.0, "content": ". Then a bijection", "type": "text"}], "index": 7}, {"bbox": [71, 195, 543, 218], "spans": [{"bbox": [71, 198, 214, 216], "score": 0.92, "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 143}, {"bbox": [214, 195, 543, 218], "score": 1.0, "content": " defines an isomorphism of fusion rings iff there exists some", "type": "text"}], "index": 8}, {"bbox": [68, 212, 544, 236], "spans": [{"bbox": [68, 212, 119, 236], "score": 1.0, "content": "bijection ", "type": "text"}, {"bbox": [120, 216, 261, 233], "score": 0.92, "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 17, "width": 141}, {"bbox": [261, 212, 317, 236], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [317, 216, 392, 232], "score": 0.94, "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [392, 212, 433, 236], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [433, 215, 518, 233], "score": 0.91, "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 18, "width": 85}, {"bbox": [518, 212, 544, 236], "score": 1.0, "content": ". In", "type": "text"}], "index": 9}, {"bbox": [70, 232, 541, 247], "spans": [{"bbox": [70, 232, 205, 247], "score": 1.0, "content": "particular, a permutation ", "type": "text"}, {"bbox": [206, 238, 213, 243], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 232, 344, 247], "score": 1.0, "content": " is a fusion-symmetry iff", "type": "text"}, {"bbox": [344, 233, 377, 246], "score": 0.92, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [377, 232, 409, 247], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [409, 234, 418, 244], "score": 0.83, "content": "S", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [418, 232, 524, 247], "score": 1.0, "content": "-symmetry for some ", "type": "text"}, {"bbox": [525, 234, 536, 243], "score": 0.87, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [536, 232, 541, 247], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "text", "bbox": [70, 250, 558, 344], "lines": [{"bbox": [68, 250, 542, 273], "spans": [{"bbox": [68, 250, 185, 273], "score": 1.0, "content": "Proof. The equality ", "type": "text"}, {"bbox": [185, 256, 263, 270], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [263, 250, 378, 273], "score": 1.0, "content": " means that, for each ", "type": "text"}, {"bbox": [378, 256, 387, 267], "score": 0.75, "content": "\\mu", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [387, 250, 497, 273], "score": 1.0, "content": ", the column vectors ", "type": "text"}, {"bbox": [498, 254, 542, 269], "score": 0.9, "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "type": "inline_equation", "height": 15, "width": 44}], "index": 11}, {"bbox": [71, 268, 559, 290], "spans": [{"bbox": [71, 270, 105, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "type": "inline_equation", "height": 16, "width": 34}, {"bbox": [105, 268, 377, 290], "score": 1.0, "content": " are simultaneous eigenvectors for the fusion matrices", "type": "text"}, {"bbox": [378, 272, 394, 284], "score": 0.89, "content": "N_{\\lambda}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [394, 268, 486, 290], "score": 1.0, "content": ", with eigenvalues", "type": "text"}, {"bbox": [487, 269, 555, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "type": "inline_equation", "height": 17, "width": 68}, {"bbox": [555, 268, 559, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 12}, {"bbox": [70, 285, 542, 301], "spans": [{"bbox": [70, 285, 542, 301], "score": 1.0, "content": "I t is easy to see from Verlinde\u2019s formula (1.1b) that any simultaneous eigenvec t or for a ll", "type": "text"}], "index": 13}, {"bbox": [71, 300, 542, 315], "spans": [{"bbox": [71, 300, 402, 315], "score": 1.0, "content": "fusion matrices must be a scalar multiple of some column of ", "type": "text"}, {"bbox": [402, 302, 410, 311], "score": 0.85, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [411, 300, 542, 315], "score": 1.0, "content": ". Thus there must be a", "type": "text"}], "index": 14}, {"bbox": [68, 312, 543, 335], "spans": [{"bbox": [68, 312, 140, 335], "score": 1.0, "content": "permutation ", "type": "text"}, {"bbox": [141, 317, 155, 327], "score": 0.9, "content": "\\pi^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [155, 312, 174, 335], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [174, 314, 223, 331], "score": 0.95, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [224, 312, 293, 335], "score": 1.0, "content": " and scalars ", "type": "text"}, {"bbox": [293, 317, 317, 330], "score": 0.94, "content": "\\alpha(\\mu)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [318, 312, 378, 335], "score": 1.0, "content": " such that", "type": "text"}, {"bbox": [378, 315, 491, 330], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [492, 312, 543, 335], "score": 1.0, "content": ". Taking", "type": "text"}], "index": 15}, {"bbox": [71, 331, 412, 345], "spans": [{"bbox": [71, 333, 100, 342], "score": 0.91, "content": "\\nu=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [100, 331, 137, 345], "score": 1.0, "content": " forces ", "type": "text"}, {"bbox": [137, 332, 184, 344], "score": 0.94, "content": "\\alpha(\\mu)>0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [184, 331, 325, 345], "score": 1.0, "content": ", and then unitarity forces ", "type": "text"}, {"bbox": [325, 332, 371, 344], "score": 0.94, "content": "\\alpha(\\mu)=1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [372, 331, 377, 345], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [401, 332, 412, 344], "score": 0.9251790046691895, "content": "\u25a0", "type": "text"}], "index": 16}], "index": 13.5}, {"type": "text", "bbox": [70, 349, 541, 391], "lines": [{"bbox": [92, 351, 541, 366], "spans": [{"bbox": [92, 351, 118, 366], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [119, 357, 126, 362], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [126, 351, 294, 366], "score": 1.0, "content": " be any isomorphism, and let ", "type": "text"}, {"bbox": [294, 353, 305, 362], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 351, 470, 366], "score": 1.0, "content": " be as in the Lemma. Then ", "type": "text"}, {"bbox": [470, 353, 481, 362], "score": 0.89, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 351, 541, 366], "score": 1.0, "content": " is also an", "type": "text"}], "index": 17}, {"bbox": [71, 367, 538, 381], "spans": [{"bbox": [71, 367, 175, 381], "score": 1.0, "content": "isomorphism, with ", "type": "text"}, {"bbox": [175, 367, 227, 380], "score": 0.94, "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [228, 367, 410, 381], "score": 1.0, "content": ". Equation (2.2b) implies for all ", "type": "text"}, {"bbox": [411, 367, 453, 380], "score": 0.91, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [454, 367, 538, 381], "score": 1.0, "content": " and all simple-", "type": "text"}], "index": 18}, {"bbox": [71, 382, 154, 393], "spans": [{"bbox": [71, 382, 117, 393], "score": 1.0, "content": "currents ", "type": "text"}, {"bbox": [117, 383, 123, 393], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [124, 382, 154, 393], "score": 1.0, "content": ", that", "type": "text"}], "index": 19}], "index": 18}, {"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 414, 541, 458], "lines": [{"bbox": [94, 416, 539, 432], "spans": [{"bbox": [94, 416, 528, 432], "score": 1.0, "content": "Another quick consequence of the Lemma is that for any Galois automorphism ", "type": "text"}, {"bbox": [528, 422, 539, 429], "score": 0.86, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 7, "width": 11}], "index": 21}, {"bbox": [69, 430, 542, 447], "spans": [{"bbox": [69, 430, 164, 447], "score": 1.0, "content": "and isomorphism ", "type": "text"}, {"bbox": [165, 436, 172, 442], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [173, 430, 225, 447], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [225, 432, 300, 444], "score": 0.94, "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [301, 430, 327, 447], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 430, 416, 445], "score": 0.94, "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "type": "inline_equation", "height": 15, "width": 88}, {"bbox": [416, 430, 542, 447], "score": 1.0, "content": ". To see this, apply the", "type": "text"}], "index": 22}, {"bbox": [71, 445, 243, 460], "spans": [{"bbox": [71, 445, 149, 460], "score": 1.0, "content": "invertibility of ", "type": "text"}, {"bbox": [150, 447, 158, 456], "score": 0.91, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [158, 445, 243, 460], "score": 1.0, "content": " to the equation", "type": "text"}], "index": 23}], "index": 22}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [69, 496, 542, 540], "lines": [{"bbox": [95, 498, 542, 514], "spans": [{"bbox": [95, 498, 542, 514], "score": 1.0, "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "type": "text"}], "index": 25}, {"bbox": [70, 511, 542, 529], "spans": [{"bbox": [70, 511, 108, 529], "score": 1.0, "content": "subset ", "type": "text"}, {"bbox": [108, 514, 199, 527], "score": 0.93, "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [200, 511, 217, 529], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [218, 515, 233, 526], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [234, 511, 325, 529], "score": 1.0, "content": " which generates ", "type": "text"}, {"bbox": [325, 513, 369, 527], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 511, 542, 529], "score": 1.0, "content": " as a ring. Diagonalising, this is", "type": "text"}], "index": 26}, {"bbox": [72, 527, 529, 543], "spans": [{"bbox": [72, 527, 270, 543], "score": 1.0, "content": "equivalent to requiring that there are ", "type": "text"}, {"bbox": [270, 532, 281, 538], "score": 0.78, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [281, 527, 396, 543], "score": 1.0, "content": "-variable polynomials ", "type": "text"}, {"bbox": [396, 528, 474, 541], "score": 0.93, "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [474, 527, 529, 543], "score": 1.0, "content": " such that", "type": "text"}], "index": 27}], "index": 26}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [69, 592, 541, 621], "lines": [{"bbox": [70, 594, 540, 609], "spans": [{"bbox": [70, 594, 93, 609], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [93, 595, 126, 608], "score": 0.94, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [126, 594, 163, 609], "score": 1.0, "content": " be an ", "type": "text"}, {"bbox": [164, 596, 172, 605], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [172, 594, 378, 609], "score": 1.0, "content": "-symmetry, and suppose we know that ", "type": "text"}, {"bbox": [379, 596, 418, 607], "score": 0.84, "content": "\\pi\\gamma=\\gamma", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [419, 594, 457, 609], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [457, 599, 465, 607], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [465, 594, 540, 609], "score": 1.0, "content": " in the fusion-", "type": "text"}], "index": 29}, {"bbox": [70, 608, 254, 624], "spans": [{"bbox": [70, 608, 124, 624], "score": 1.0, "content": "generator ", "type": "text"}, {"bbox": [124, 610, 132, 619], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [132, 608, 212, 624], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [213, 610, 250, 622], "score": 0.93, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [250, 608, 254, 624], "score": 1.0, "content": ",", "type": "text"}], "index": 30}], "index": 29.5}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [69, 672, 207, 686], "lines": [{"bbox": [71, 674, 207, 687], "spans": [{"bbox": [71, 674, 106, 687], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 676, 144, 687], "score": 0.93, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [144, 674, 165, 687], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [165, 676, 203, 685], "score": 0.93, "content": "\\pi\\lambda=\\lambda", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [203, 674, 207, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [70, 687, 541, 715], "lines": [{"bbox": [95, 689, 541, 703], "spans": [{"bbox": [95, 689, 541, 703], "score": 1.0, "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "type": "text"}], "index": 33}, {"bbox": [72, 703, 540, 717], "spans": [{"bbox": [72, 703, 540, 717], "score": 1.0, "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "type": "text"}], "index": 34}], "index": 33.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 392, 85], "lines": [{"bbox": [70, 74, 390, 87], "spans": [{"bbox": [70, 74, 139, 87], "score": 1.0, "content": "For instance ", "type": "text"}, {"bbox": [139, 79, 146, 84], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [147, 74, 207, 87], "score": 1.0, "content": " must send ", "type": "text"}, {"bbox": [208, 75, 216, 84], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [216, 74, 298, 87], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [298, 75, 323, 87], "score": 0.94, "content": "\\pi(J)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [324, 74, 390, 87], "score": 1.0, "content": "-fixed-points.", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 74, 390, 87]}, {"type": "text", "bbox": [70, 86, 542, 160], "lines": [{"bbox": [95, 88, 541, 102], "spans": [{"bbox": [95, 88, 309, 102], "score": 1.0, "content": "More generally, a fusion-homomorphism ", "type": "text"}, {"bbox": [309, 93, 316, 99], "score": 0.85, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [317, 88, 541, 102], "score": 1.0, "content": " is defined in the obvious algebraic way. It", "type": "text"}], "index": 1}, {"bbox": [70, 102, 541, 117], "spans": [{"bbox": [70, 102, 212, 117], "score": 1.0, "content": "turns out that for such a ", "type": "text"}, {"bbox": [213, 108, 220, 113], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [220, 102, 228, 117], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [228, 104, 278, 115], "score": 0.91, "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [278, 102, 298, 117], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [298, 104, 341, 115], "score": 0.93, "content": "\\mu\\,=\\,J\\lambda", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [341, 102, 476, 117], "score": 1.0, "content": " for some simple-current ", "type": "text"}, {"bbox": [477, 105, 484, 113], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [485, 102, 541, 117], "score": 1.0, "content": " for which", "type": "text"}], "index": 2}, {"bbox": [71, 116, 542, 132], "spans": [{"bbox": [71, 117, 124, 131], "score": 0.94, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [124, 116, 187, 132], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [187, 117, 240, 131], "score": 0.95, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [241, 116, 404, 132], "score": 1.0, "content": " is possible only if there are no ", "type": "text"}, {"bbox": [404, 119, 412, 127], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [413, 116, 520, 132], "score": 1.0, "content": "-fixed-points. When ", "type": "text"}, {"bbox": [520, 122, 528, 127], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [528, 116, 542, 132], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [69, 131, 542, 149], "spans": [{"bbox": [69, 131, 409, 149], "score": 1.0, "content": "one-to-one (e.g. when there are no nontrivial simple-currents in ", "type": "text"}, {"bbox": [410, 131, 463, 148], "score": 0.93, "content": "P_{+}^{k}(X_{r}^{(1)}))", "type": "inline_equation", "height": 17, "width": 53}, {"bbox": [463, 131, 498, 149], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [498, 138, 506, 144], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [506, 131, 542, 149], "score": 1.0, "content": " obeys", "type": "text"}], "index": 4}, {"bbox": [72, 148, 366, 162], "spans": [{"bbox": [72, 148, 366, 162], "score": 1.0, "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "type": "text"}], "index": 5}], "index": 3, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 88, 542, 162]}, {"type": "text", "bbox": [95, 160, 420, 174], "lines": [{"bbox": [95, 162, 419, 176], "spans": [{"bbox": [95, 162, 419, 176], "score": 1.0, "content": "The key to finding fusion-symmetries is the following Lemma.", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [95, 162, 419, 176]}, {"type": "text", "bbox": [70, 180, 543, 246], "lines": [{"bbox": [91, 178, 543, 201], "spans": [{"bbox": [91, 178, 194, 201], "score": 1.0, "content": "Lemma 2.2. Let", "type": "text"}, {"bbox": [194, 183, 203, 196], "score": 0.87, "content": "\\widetilde{S}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [203, 178, 376, 201], "score": 1.0, "content": " be the Kac-Peterson matrix for ", "type": "text"}, {"bbox": [376, 181, 400, 197], "score": 0.9, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 16, "width": 24}, {"bbox": [400, 178, 431, 201], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [431, 186, 443, 196], "score": 0.46, "content": "m", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [443, 178, 543, 201], "score": 1.0, "content": ". Then a bijection", "type": "text"}], "index": 7}, {"bbox": [71, 195, 543, 218], "spans": [{"bbox": [71, 198, 214, 216], "score": 0.92, "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 143}, {"bbox": [214, 195, 543, 218], "score": 1.0, "content": " defines an isomorphism of fusion rings iff there exists some", "type": "text"}], "index": 8}, {"bbox": [68, 212, 544, 236], "spans": [{"bbox": [68, 212, 119, 236], "score": 1.0, "content": "bijection ", "type": "text"}, {"bbox": [120, 216, 261, 233], "score": 0.92, "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 17, "width": 141}, {"bbox": [261, 212, 317, 236], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [317, 216, 392, 232], "score": 0.94, "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [392, 212, 433, 236], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [433, 215, 518, 233], "score": 0.91, "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 18, "width": 85}, {"bbox": [518, 212, 544, 236], "score": 1.0, "content": ". In", "type": "text"}], "index": 9}, {"bbox": [70, 232, 541, 247], "spans": [{"bbox": [70, 232, 205, 247], "score": 1.0, "content": "particular, a permutation ", "type": "text"}, {"bbox": [206, 238, 213, 243], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 232, 344, 247], "score": 1.0, "content": " is a fusion-symmetry iff", "type": "text"}, {"bbox": [344, 233, 377, 246], "score": 0.92, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [377, 232, 409, 247], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [409, 234, 418, 244], "score": 0.83, "content": "S", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [418, 232, 524, 247], "score": 1.0, "content": "-symmetry for some ", "type": "text"}, {"bbox": [525, 234, 536, 243], "score": 0.87, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [536, 232, 541, 247], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [68, 178, 544, 247]}, {"type": "text", "bbox": [70, 250, 558, 344], "lines": [{"bbox": [68, 250, 542, 273], "spans": [{"bbox": [68, 250, 185, 273], "score": 1.0, "content": "Proof. The equality ", "type": "text"}, {"bbox": [185, 256, 263, 270], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [263, 250, 378, 273], "score": 1.0, "content": " means that, for each ", "type": "text"}, {"bbox": [378, 256, 387, 267], "score": 0.75, "content": "\\mu", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [387, 250, 497, 273], "score": 1.0, "content": ", the column vectors ", "type": "text"}, {"bbox": [498, 254, 542, 269], "score": 0.9, "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "type": "inline_equation", "height": 15, "width": 44}], "index": 11}, {"bbox": [71, 268, 559, 290], "spans": [{"bbox": [71, 270, 105, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "type": "inline_equation", "height": 16, "width": 34}, {"bbox": [105, 268, 377, 290], "score": 1.0, "content": " are simultaneous eigenvectors for the fusion matrices", "type": "text"}, {"bbox": [378, 272, 394, 284], "score": 0.89, "content": "N_{\\lambda}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [394, 268, 486, 290], "score": 1.0, "content": ", with eigenvalues", "type": "text"}, {"bbox": [487, 269, 555, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "type": "inline_equation", "height": 17, "width": 68}, {"bbox": [555, 268, 559, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 12}, {"bbox": [70, 285, 542, 301], "spans": [{"bbox": [70, 285, 542, 301], "score": 1.0, "content": "I t is easy to see from Verlinde\u2019s formula (1.1b) that any simultaneous eigenvec t or for a ll", "type": "text"}], "index": 13}, {"bbox": [71, 300, 542, 315], "spans": [{"bbox": [71, 300, 402, 315], "score": 1.0, "content": "fusion matrices must be a scalar multiple of some column of ", "type": "text"}, {"bbox": [402, 302, 410, 311], "score": 0.85, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [411, 300, 542, 315], "score": 1.0, "content": ". Thus there must be a", "type": "text"}], "index": 14}, {"bbox": [68, 312, 543, 335], "spans": [{"bbox": [68, 312, 140, 335], "score": 1.0, "content": "permutation ", "type": "text"}, {"bbox": [141, 317, 155, 327], "score": 0.9, "content": "\\pi^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [155, 312, 174, 335], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [174, 314, 223, 331], "score": 0.95, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [224, 312, 293, 335], "score": 1.0, "content": " and scalars ", "type": "text"}, {"bbox": [293, 317, 317, 330], "score": 0.94, "content": "\\alpha(\\mu)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [318, 312, 378, 335], "score": 1.0, "content": " such that", "type": "text"}, {"bbox": [378, 315, 491, 330], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [492, 312, 543, 335], "score": 1.0, "content": ". Taking", "type": "text"}], "index": 15}, {"bbox": [71, 331, 412, 345], "spans": [{"bbox": [71, 333, 100, 342], "score": 0.91, "content": "\\nu=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [100, 331, 137, 345], "score": 1.0, "content": " forces ", "type": "text"}, {"bbox": [137, 332, 184, 344], "score": 0.94, "content": "\\alpha(\\mu)>0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [184, 331, 325, 345], "score": 1.0, "content": ", and then unitarity forces ", "type": "text"}, {"bbox": [325, 332, 371, 344], "score": 0.94, "content": "\\alpha(\\mu)=1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [372, 331, 377, 345], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [401, 332, 412, 344], "score": 0.9251790046691895, "content": "\u25a0", "type": "text"}], "index": 16}], "index": 13.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [68, 250, 559, 345]}, {"type": "text", "bbox": [70, 349, 541, 391], "lines": [{"bbox": [92, 351, 541, 366], "spans": [{"bbox": [92, 351, 118, 366], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [119, 357, 126, 362], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [126, 351, 294, 366], "score": 1.0, "content": " be any isomorphism, and let ", "type": "text"}, {"bbox": [294, 353, 305, 362], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 351, 470, 366], "score": 1.0, "content": " be as in the Lemma. Then ", "type": "text"}, {"bbox": [470, 353, 481, 362], "score": 0.89, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 351, 541, 366], "score": 1.0, "content": " is also an", "type": "text"}], "index": 17}, {"bbox": [71, 367, 538, 381], "spans": [{"bbox": [71, 367, 175, 381], "score": 1.0, "content": "isomorphism, with ", "type": "text"}, {"bbox": [175, 367, 227, 380], "score": 0.94, "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [228, 367, 410, 381], "score": 1.0, "content": ". Equation (2.2b) implies for all ", "type": "text"}, {"bbox": [411, 367, 453, 380], "score": 0.91, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [454, 367, 538, 381], "score": 1.0, "content": " and all simple-", "type": "text"}], "index": 18}, {"bbox": [71, 382, 154, 393], "spans": [{"bbox": [71, 382, 117, 393], "score": 1.0, "content": "currents ", "type": "text"}, {"bbox": [117, 383, 123, 393], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [124, 382, 154, 393], "score": 1.0, "content": ", that", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [71, 351, 541, 393]}, {"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 414, 541, 458], "lines": [{"bbox": [94, 416, 539, 432], "spans": [{"bbox": [94, 416, 528, 432], "score": 1.0, "content": "Another quick consequence of the Lemma is that for any Galois automorphism ", "type": "text"}, {"bbox": [528, 422, 539, 429], "score": 0.86, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 7, "width": 11}], "index": 21}, {"bbox": [69, 430, 542, 447], "spans": [{"bbox": [69, 430, 164, 447], "score": 1.0, "content": "and isomorphism ", "type": "text"}, {"bbox": [165, 436, 172, 442], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [173, 430, 225, 447], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [225, 432, 300, 444], "score": 0.94, "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [301, 430, 327, 447], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 430, 416, 445], "score": 0.94, "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "type": "inline_equation", "height": 15, "width": 88}, {"bbox": [416, 430, 542, 447], "score": 1.0, "content": ". To see this, apply the", "type": "text"}], "index": 22}, {"bbox": [71, 445, 243, 460], "spans": [{"bbox": [71, 445, 149, 460], "score": 1.0, "content": "invertibility of ", "type": "text"}, {"bbox": [150, 447, 158, 456], "score": 0.91, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [158, 445, 243, 460], "score": 1.0, "content": " to the equation", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 416, 542, 460]}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 496, 542, 540], "lines": [{"bbox": [95, 498, 542, 514], "spans": [{"bbox": [95, 498, 542, 514], "score": 1.0, "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "type": "text"}], "index": 25}, {"bbox": [70, 511, 542, 529], "spans": [{"bbox": [70, 511, 108, 529], "score": 1.0, "content": "subset ", "type": "text"}, {"bbox": [108, 514, 199, 527], "score": 0.93, "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [200, 511, 217, 529], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [218, 515, 233, 526], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [234, 511, 325, 529], "score": 1.0, "content": " which generates ", "type": "text"}, {"bbox": [325, 513, 369, 527], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 511, 542, 529], "score": 1.0, "content": " as a ring. Diagonalising, this is", "type": "text"}], "index": 26}, {"bbox": [72, 527, 529, 543], "spans": [{"bbox": [72, 527, 270, 543], "score": 1.0, "content": "equivalent to requiring that there are ", "type": "text"}, {"bbox": [270, 532, 281, 538], "score": 0.78, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [281, 527, 396, 543], "score": 1.0, "content": "-variable polynomials ", "type": "text"}, {"bbox": [396, 528, 474, 541], "score": 0.93, "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [474, 527, 529, 543], "score": 1.0, "content": " such that", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 498, 542, 543]}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 592, 541, 621], "lines": [{"bbox": [70, 594, 540, 609], "spans": [{"bbox": [70, 594, 93, 609], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [93, 595, 126, 608], "score": 0.94, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [126, 594, 163, 609], "score": 1.0, "content": " be an ", "type": "text"}, {"bbox": [164, 596, 172, 605], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [172, 594, 378, 609], "score": 1.0, "content": "-symmetry, and suppose we know that ", "type": "text"}, {"bbox": [379, 596, 418, 607], "score": 0.84, "content": "\\pi\\gamma=\\gamma", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [419, 594, 457, 609], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [457, 599, 465, 607], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [465, 594, 540, 609], "score": 1.0, "content": " in the fusion-", "type": "text"}], "index": 29}, {"bbox": [70, 608, 254, 624], "spans": [{"bbox": [70, 608, 124, 624], "score": 1.0, "content": "generator ", "type": "text"}, {"bbox": [124, 610, 132, 619], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [132, 608, 212, 624], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [213, 610, 250, 622], "score": 0.93, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [250, 608, 254, 624], "score": 1.0, "content": ",", "type": "text"}], "index": 30}], "index": 29.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 594, 540, 624]}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 672, 207, 686], "lines": [{"bbox": [71, 674, 207, 687], "spans": [{"bbox": [71, 674, 106, 687], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 676, 144, 687], "score": 0.93, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [144, 674, 165, 687], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [165, 676, 203, 685], "score": 0.93, "content": "\\pi\\lambda=\\lambda", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [203, 674, 207, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [71, 674, 207, 687]}, {"type": "text", "bbox": [70, 687, 541, 715], "lines": [{"bbox": [95, 689, 541, 703], "spans": [{"bbox": [95, 689, 541, 703], "score": 1.0, "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "type": "text"}], "index": 33}, {"bbox": [72, 703, 540, 717], "spans": [{"bbox": [72, 703, 540, 717], "score": 1.0, "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "type": "text"}], "index": 34}, {"bbox": [71, 71, 542, 91], "spans": [{"bbox": [71, 75, 90, 88], "score": 0.84, "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "type": "inline_equation", "height": 13, "width": 19, "cross_page": true}, {"bbox": [90, 71, 111, 91], "score": 1.0, "content": " for ", "type": "text", "cross_page": true}, {"bbox": [112, 75, 127, 86], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [128, 71, 462, 91], "score": 1.0, "content": " can be written as a polynomial in the fundamental characters ", "type": "text", "cross_page": true}, {"bbox": [462, 74, 537, 87], "score": 0.87, "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "type": "inline_equation", "height": 13, "width": 75, "cross_page": true}, {"bbox": [537, 71, 542, 91], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [69, 88, 540, 106], "spans": [{"bbox": [69, 88, 212, 106], "score": 1.0, "content": "we know from (2.1b) that ", "type": "text", "cross_page": true}, {"bbox": [212, 92, 303, 105], "score": 0.93, "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 91, "cross_page": true}, {"bbox": [304, 88, 438, 106], "score": 1.0, "content": " is a fusion-generator for ", "type": "text", "cross_page": true}, {"bbox": [438, 88, 462, 103], "score": 0.92, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24, "cross_page": true}, {"bbox": [463, 88, 532, 106], "score": 1.0, "content": "at any level ", "type": "text", "cross_page": true}, {"bbox": [532, 93, 540, 102], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}], "index": 1}, {"bbox": [70, 105, 541, 120], "spans": [{"bbox": [70, 105, 185, 120], "score": 1.0, "content": "sufficiently large that ", "type": "text", "cross_page": true}, {"bbox": [186, 108, 201, 119], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [202, 105, 268, 120], "score": 1.0, "content": " contains all ", "type": "text", "cross_page": true}, {"bbox": [268, 107, 281, 118], "score": 0.9, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [281, 105, 412, 120], "score": 1.0, "content": " (in other words, for any ", "type": "text", "cross_page": true}, {"bbox": [412, 105, 478, 119], "score": 0.91, "content": "k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}", "type": "inline_equation", "height": 14, "width": 66, "cross_page": true}, {"bbox": [479, 105, 541, 120], "score": 1.0, "content": "). In fact, it", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [70, 119, 541, 135], "spans": [{"bbox": [70, 119, 362, 135], "score": 1.0, "content": "is easy to show [18] that a fusion-generator valid for any ", "type": "text", "cross_page": true}, {"bbox": [362, 122, 385, 134], "score": 0.93, "content": "X_{r,k}", "type": "inline_equation", "height": 12, "width": 23, "cross_page": true}, {"bbox": [385, 119, 399, 135], "score": 1.0, "content": " is ", "type": "text", "cross_page": true}, {"bbox": [400, 120, 492, 133], "score": 0.92, "content": "\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}", "type": "inline_equation", "height": 13, "width": 92, "cross_page": true}, {"bbox": [492, 119, 541, 135], "score": 1.0, "content": ". Smaller", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [70, 133, 541, 149], "spans": [{"bbox": [70, 133, 310, 149], "score": 1.0, "content": "fusion-generators usually exist \u2014 for example ", "type": "text", "cross_page": true}, {"bbox": [310, 135, 336, 147], "score": 0.94, "content": "\\{\\Lambda_{1}\\}", "type": "inline_equation", "height": 12, "width": 26, "cross_page": true}, {"bbox": [336, 133, 465, 149], "score": 1.0, "content": " is a fusion-generator for ", "type": "text", "cross_page": true}, {"bbox": [465, 135, 487, 148], "score": 0.91, "content": "A_{8,k}", "type": "inline_equation", "height": 13, "width": 22, "cross_page": true}, {"bbox": [488, 133, 541, 149], "score": 1.0, "content": " whenever", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [71, 149, 215, 163], "spans": [{"bbox": [71, 150, 78, 159], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7, "cross_page": true}, {"bbox": [78, 149, 215, 163], "score": 1.0, "content": " is even and coprime to 3.", "type": "text", "cross_page": true}], "index": 5}], "index": 33.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [72, 689, 541, 717]}]}
0002044v1	11	valid for all $$1\leq i<r-2$$ and $$k>2$$ . We also will use the character formula (2.1b) where $$\lambda^{+}(\ell)\,=\,(\lambda+\rho)(\ell)$$ and the orthonormal components $$\lambda(\ell)$$ are defined by $$\lambda(\ell)\,=$$ $$\begin{array}{r}{\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{\lambda_{r}-\lambda_{r-1}}{2}}\end{array}$$ . The simple-current automorphisms are as follows, and depend on whether $$r$$ and $$k$$ are even or odd. When $$r$$ is odd, the group of simple-currents is generated by $$J_{s}$$ . If in addition $$k$$ is odd, there will be only two simple-current automorphisms: $$\pi=\pi^{\prime}=\pi[a]=J_{s}^{4a\cup_{s}}$$ for $$a\in\{0,2\}$$ . If instead $$k$$ is even, there will be four simple-current automorphisms: $$\pi=\pi[a]$$ and $$\pi^{\prime}=\pi[a k-a]$$ for $$0\leq a\leq3$$ . When $$k\equiv2$$ (mod 4), these form the group $$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$ , otherwise when $$4\|k$$ the group is $$\mathbb{Z}_{4}$$ . When $$r$$ is even, the simple-currents are generated by both $$J_{v}$$ and $$J_{s}$$ . If in addition $$k$$ is even, we have 16 simple-current automorphisms: for any $$a,b,c,d\in\{0,1\}$$ , forming a group isomorphic to $$\mathbb{Z}_{2}^{4}$$ . This notation means When $$k$$ is odd, we will have six simple-current automorphisms: where $$a={\frac{r}{2}}$$ or $$d=0$$ , and where $$b=1$$ or $$d=1$$ . The corresponding permutation of $$P_{+}$$ is still given by (3.5). Again, for these $$r,k$$ , these are the values of $$a,b,c,d$$ for which (3.5) is invertible. For $$k$$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $$\mathfrak{S}_{3}$$ when 4 divides $$r$$ , and to $$\mathbb{Z}_{6}$$ when $$r\equiv2$$ (mod 4). For $$k=2$$ (so $$\kappa=2r$$ ), there are several Galois fusion-symmetries. In particular, write $$\lambda^{i}\,=\,\lambda^{2r-i}\,=\,\Lambda_{i}$$ for $$1\,\leq\,i\,\leq\,r\,-\,2$$ , and $$\lambda^{r\pm1}=\Lambda_{r-1}+\Lambda_{r}$$ . As with $$B_{r,2}$$ , $$\begin{array}{r}{S_{00}^{2}\,=\,\frac{1}{4\kappa}}\end{array}$$ 1 is rational so for any $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ coprime to $$2r$$ , we get a Galois fusion-symmetry $$\pi\{m\}$$ . It takes $$\lambda^{a}$$ to $$\lambda^{m a}$$ , where the superscript is taken mod $$2r$$ , and will fix $$J_{v}0$$ . Also, $$\pi\{m\}$$ will send $$J_{s}0$$ to $$J_{s}^{m}0$$ , as well as stabilise the set $$\left\{\Lambda_{r},\Lambda_{r-1},J_{v}\Lambda_{r},J_{v}\Lambda_{r-1}\right\}$$ . (In particular, put $$t=r$$ when $$r$$ is even or when $$m\equiv1$$ (mod 4), otherwise put $$t=r-1$$ ; then for any $$i,j,\,\pi\{m\}\,C_{1}^{j}J_{v}^{i}\Lambda_{r}$$ is $$C_{1}^{j}J_{v}^{i}\Lambda_{t}$$ or $$C_{1}^{j}J_{v}^{i+1}\Lambda_{t}$$ , when the Jacobi symbol $$\textstyle\left({\frac{\kappa}{m}}\right)$$ is $$\pm1$$ , respectively.) Theorem 3.D. The fusion-symmetries of $$D_{r}^{(1)}$$ for $$k\neq2$$ are all of the form $$C_{i}\,\pi$$ , where $$C_{i}$$ is a conjugation, and where $$\pi$$ is a simple-current automorphism. Similarly for	<p>valid for all $$1\leq i<r-2$$ and $$k>2$$ . We also will use the character formula (2.1b)</p> <p>where $$\lambda^{+}(\ell)\,=\,(\lambda+\rho)(\ell)$$ and the orthonormal components $$\lambda(\ell)$$ are defined by $$\lambda(\ell)\,=$$ $$\begin{array}{r}{\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{\lambda_{r}-\lambda_{r-1}}{2}}\end{array}$$ .</p> <p>The simple-current automorphisms are as follows, and depend on whether $$r$$ and $$k$$ are even or odd. When $$r$$ is odd, the group of simple-currents is generated by $$J_{s}$$ . If in addition $$k$$ is odd, there will be only two simple-current automorphisms: $$\pi=\pi^{\prime}=\pi[a]=J_{s}^{4a\cup_{s}}$$ for $$a\in\{0,2\}$$ . If instead $$k$$ is even, there will be four simple-current automorphisms: $$\pi=\pi[a]$$ and $$\pi^{\prime}=\pi[a k-a]$$ for $$0\leq a\leq3$$ . When $$k\equiv2$$ (mod 4), these form the group $$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$ , otherwise when $$4\|k$$ the group is $$\mathbb{Z}_{4}$$ .</p> <p>When $$r$$ is even, the simple-currents are generated by both $$J_{v}$$ and $$J_{s}$$ . If in addition $$k$$ is even, we have 16 simple-current automorphisms:</p> <p>for any $$a,b,c,d\in\{0,1\}$$ , forming a group isomorphic to $$\mathbb{Z}_{2}^{4}$$ . This notation means</p> <p>When $$k$$ is odd, we will have six simple-current automorphisms:</p> <p>where $$a={\frac{r}{2}}$$ or $$d=0$$ , and where $$b=1$$ or $$d=1$$ . The corresponding permutation of $$P_{+}$$ is still given by (3.5). Again, for these $$r,k$$ , these are the values of $$a,b,c,d$$ for which (3.5) is invertible. For $$k$$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $$\mathfrak{S}_{3}$$ when 4 divides $$r$$ , and to $$\mathbb{Z}_{6}$$ when $$r\equiv2$$ (mod 4).</p> <p>For $$k=2$$ (so $$\kappa=2r$$ ), there are several Galois fusion-symmetries. In particular, write $$\lambda^{i}\,=\,\lambda^{2r-i}\,=\,\Lambda_{i}$$ for $$1\,\leq\,i\,\leq\,r\,-\,2$$ , and $$\lambda^{r\pm1}=\Lambda_{r-1}+\Lambda_{r}$$ . As with $$B_{r,2}$$ , $$\begin{array}{r}{S_{00}^{2}\,=\,\frac{1}{4\kappa}}\end{array}$$ 1 is rational so for any $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ coprime to $$2r$$ , we get a Galois fusion-symmetry $$\pi\{m\}$$ . It takes $$\lambda^{a}$$ to $$\lambda^{m a}$$ , where the superscript is taken mod $$2r$$ , and will fix $$J_{v}0$$ . Also, $$\pi\{m\}$$ will send $$J_{s}0$$ to $$J_{s}^{m}0$$ , as well as stabilise the set $$\left\{\Lambda_{r},\Lambda_{r-1},J_{v}\Lambda_{r},J_{v}\Lambda_{r-1}\right\}$$ . (In particular, put $$t=r$$ when $$r$$ is even or when $$m\equiv1$$ (mod 4), otherwise put $$t=r-1$$ ; then for any $$i,j,\,\pi\{m\}\,C_{1}^{j}J_{v}^{i}\Lambda_{r}$$ is $$C_{1}^{j}J_{v}^{i}\Lambda_{t}$$ or $$C_{1}^{j}J_{v}^{i+1}\Lambda_{t}$$ , when the Jacobi symbol $$\textstyle\left({\frac{\kappa}{m}}\right)$$ is $$\pm1$$ , respectively.)</p> <p>Theorem 3.D. The fusion-symmetries of $$D_{r}^{(1)}$$ for $$k\neq2$$ are all of the form $$C_{i}\,\pi$$ , where $$C_{i}$$ is a conjugation, and where $$\pi$$ is a simple-current automorphism. Similarly for</p>	[{"type": "text", "coordinates": [70, 70, 505, 86], "content": "valid for all $$1\\leq i<r-2$$ and $$k>2$$ . We also will use the character formula (2.1b)", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [205, 99, 405, 137], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 147, 541, 178], "content": "where $$\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)$$ and the orthonormal components $$\\lambda(\\ell)$$ are defined by $$\\lambda(\\ell)\\,=$$\n$$\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 178, 542, 264], "content": "The simple-current automorphisms are as follows, and depend on whether $$r$$ and $$k$$ are\neven or odd. When $$r$$ is odd, the group of simple-currents is generated by $$J_{s}$$ . If in addition\n$$k$$ is odd, there will be only two simple-current automorphisms: $$\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}$$ for\n$$a\\in\\{0,2\\}$$ . If instead $$k$$ is even, there will be four simple-current automorphisms: $$\\pi=\\pi[a]$$\nand $$\\pi^{\\prime}=\\pi[a k-a]$$ for $$0\\leq a\\leq3$$ . When $$k\\equiv2$$ (mod 4), these form the group $$\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$$ ,\notherwise when $$4\|k$$ the group is $$\\mathbb{Z}_{4}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 264, 541, 293], "content": "When $$r$$ is even, the simple-currents are generated by both $$J_{v}$$ and $$J_{s}$$ . If in addition\n$$k$$ is even, we have 16 simple-current automorphisms:", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [195, 305, 415, 337], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [69, 346, 495, 362], "content": "for any $$a,b,c,d\\in\\{0,1\\}$$ , forming a group isomorphic to $$\\mathbb{Z}_{2}^{4}$$ . This notation means", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [164, 375, 448, 406], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [70, 415, 407, 430], "content": "When $$k$$ is odd, we will have six simple-current automorphisms:", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [126, 441, 490, 510], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [69, 516, 541, 574], "content": "where $$a={\\frac{r}{2}}$$ or $$d=0$$ , and where $$b=1$$ or $$d=1$$ . The corresponding permutation of $$P_{+}$$\nis still given by (3.5). Again, for these $$r,k$$ , these are the values of $$a,b,c,d$$ for which (3.5)\nis invertible. For $$k$$ odd, the group of simple-current automorphisms is isomorphic to the\nsymmetric group $$\\mathfrak{S}_{3}$$ when 4 divides $$r$$ , and to $$\\mathbb{Z}_{6}$$ when $$r\\equiv2$$ (mod 4).", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 574, 542, 678], "content": "For $$k=2$$ (so $$\\kappa=2r$$ ), there are several Galois fusion-symmetries. In particular, write\n$$\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}$$ for $$1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2$$ , and $$\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}$$ . As with $$B_{r,2}$$ , $$\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}$$ 1 is\nrational so for any $${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$$ coprime to $$2r$$ , we get a Galois fusion-symmetry $$\\pi\\{m\\}$$ . It takes $$\\lambda^{a}$$ to\n$$\\lambda^{m a}$$ , where the superscript is taken mod $$2r$$ , and will fix $$J_{v}0$$ . Also, $$\\pi\\{m\\}$$ will send $$J_{s}0$$ to\n$$J_{s}^{m}0$$ , as well as stabilise the set $$\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}$$ . (In particular, put $$t=r$$ when\n$$r$$ is even or when $$m\\equiv1$$ (mod 4), otherwise put $$t=r-1$$ ; then for any $$i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}$$\nis $$C_{1}^{j}J_{v}^{i}\\Lambda_{t}$$ or $$C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}$$ , when the Jacobi symbol $$\\textstyle\\left({\\frac{\\kappa}{m}}\\right)$$ is $$\\pm1$$ , respectively.)", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [70, 684, 541, 716], "content": "Theorem 3.D. The fusion-symmetries of $$D_{r}^{(1)}$$ for $$k\\neq2$$ are all of the form $$C_{i}\\,\\pi$$ ,\nwhere $$C_{i}$$ is a conjugation, and where $$\\pi$$ is a simple-current automorphism. Similarly for", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [71, 73, 135, 88], "content": "valid for all ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [135, 75, 204, 86], "content": "1\\leq i<r-2", "score": 0.88, "index": 2}, {"type": "text", "coordinates": [204, 73, 230, 88], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [230, 75, 259, 85], "content": "k>2", "score": 0.88, "index": 4}, {"type": "text", "coordinates": [259, 73, 504, 88], "content": ". We also will use the character formula (2.1b)", "score": 1.0, "index": 5}, {"type": "interline_equation", "coordinates": [205, 99, 405, 137], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "score": 0.94, "index": 6}, {"type": "text", "coordinates": [70, 149, 106, 166], "content": "where ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [106, 150, 209, 163], "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "score": 0.92, "index": 8}, {"type": "text", "coordinates": [209, 149, 396, 166], "content": " and the orthonormal components ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [396, 151, 418, 163], "content": "\\lambda(\\ell)", "score": 0.93, "index": 10}, {"type": "text", "coordinates": [418, 149, 503, 166], "content": " are defined by ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [503, 150, 541, 164], "content": "\\lambda(\\ell)\\,=", "score": 0.91, "index": 12}, {"type": "inline_equation", "coordinates": [71, 163, 171, 180], "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [171, 162, 175, 180], "content": ".", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [76, 178, 481, 196], "content": "The simple-current automorphisms are as follows, and depend on whether ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [482, 185, 487, 191], "content": "r", "score": 0.86, "index": 16}, {"type": "text", "coordinates": [488, 178, 513, 196], "content": " and ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [513, 182, 520, 191], "content": "k", "score": 0.89, "index": 18}, {"type": "text", "coordinates": [520, 178, 542, 196], "content": " are", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [69, 194, 173, 209], "content": "even or odd. When", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [174, 199, 180, 205], "content": "r", "score": 0.67, "index": 21}, {"type": "text", "coordinates": [180, 194, 451, 209], "content": " is odd, the group of simple-currents is generated by ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [451, 196, 463, 207], "content": "J_{s}", "score": 0.9, "index": 23}, {"type": "text", "coordinates": [464, 194, 541, 209], "content": ". If in addition", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [71, 210, 78, 219], "content": "k", "score": 0.86, "index": 25}, {"type": "text", "coordinates": [78, 207, 404, 224], "content": " is odd, there will be only two simple-current automorphisms: ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [405, 209, 521, 222], "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "score": 0.93, "index": 27}, {"type": "text", "coordinates": [521, 207, 543, 224], "content": "for", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [71, 223, 122, 237], "content": "a\\in\\{0,2\\}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [122, 222, 182, 239], "content": ". If instead ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [182, 224, 190, 234], "content": "k", "score": 0.84, "index": 31}, {"type": "text", "coordinates": [190, 222, 495, 239], "content": " is even, there will be four simple-current automorphisms: ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [496, 224, 540, 236], "content": "\\pi=\\pi[a]", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [70, 236, 95, 254], "content": "and ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [95, 238, 172, 251], "content": "\\pi^{\\prime}=\\pi[a k-a]", "score": 0.93, "index": 35}, {"type": "text", "coordinates": [172, 236, 194, 254], "content": " for ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [195, 238, 248, 250], "content": "0\\leq a\\leq3", "score": 0.85, "index": 37}, {"type": "text", "coordinates": [249, 236, 293, 254], "content": ". When ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [294, 239, 324, 248], "content": "k\\equiv2", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [324, 236, 493, 254], "content": " (mod 4), these form the group ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [494, 239, 536, 250], "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [536, 236, 541, 254], "content": ",", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [70, 250, 155, 268], "content": "otherwise when ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [155, 253, 172, 265], "content": "4\|k", "score": 0.88, "index": 44}, {"type": "text", "coordinates": [172, 250, 241, 268], "content": " the group is ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [242, 254, 255, 264], "content": "\\mathbb{Z}_{4}", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [256, 250, 260, 268], "content": ".", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [95, 266, 130, 281], "content": "When ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [130, 271, 136, 277], "content": "r", "score": 0.87, "index": 49}, {"type": "text", "coordinates": [136, 266, 408, 281], "content": " is even, the simple-currents are generated by both ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [408, 268, 421, 279], "content": "J_{v}", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [421, 266, 448, 281], "content": " and ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [448, 268, 460, 279], "content": "J_{s}", "score": 0.91, "index": 53}, {"type": "text", "coordinates": [461, 266, 541, 281], "content": ". If in addition", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [71, 282, 78, 291], "content": "k", "score": 0.88, "index": 55}, {"type": "text", "coordinates": [78, 280, 349, 297], "content": " is even, we have 16 simple-current automorphisms:", "score": 1.0, "index": 56}, {"type": "interline_equation", "coordinates": [195, 305, 415, 337], "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [70, 348, 112, 366], "content": "for any ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [112, 351, 195, 363], "content": "a,b,c,d\\in\\{0,1\\}", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [195, 348, 365, 366], "content": ", forming a group isomorphic to ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [365, 350, 379, 363], "content": "\\mathbb{Z}_{2}^{4}", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [379, 348, 497, 366], "content": ". This notation means", "score": 1.0, "index": 62}, {"type": "interline_equation", "coordinates": [164, 375, 448, 406], "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [71, 417, 106, 432], "content": "When ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [106, 420, 113, 429], "content": "k", "score": 0.89, "index": 65}, {"type": "text", "coordinates": [113, 417, 405, 432], "content": " is odd, we will have six simple-current automorphisms:", "score": 1.0, "index": 66}, {"type": "interline_equation", "coordinates": [126, 441, 490, 510], "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [70, 517, 106, 536], "content": "where ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [106, 520, 137, 533], "content": "a={\\frac{r}{2}}", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [138, 517, 156, 536], "content": " or ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [156, 520, 186, 529], "content": "d=0", "score": 0.89, "index": 71}, {"type": "text", "coordinates": [187, 517, 252, 536], "content": ", and where ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [252, 519, 281, 529], "content": "b=1", "score": 0.88, "index": 73}, {"type": "text", "coordinates": [281, 517, 299, 536], "content": " or ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [299, 520, 329, 529], "content": "d=1", "score": 0.89, "index": 75}, {"type": "text", "coordinates": [330, 517, 523, 536], "content": ". The corresponding permutation of ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [523, 520, 539, 532], "content": "P_{+}", "score": 0.9, "index": 77}, {"type": "text", "coordinates": [70, 533, 274, 547], "content": "is still given by (3.5). Again, for these ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [275, 533, 293, 546], "content": "r,k", "score": 0.87, "index": 79}, {"type": "text", "coordinates": [293, 533, 419, 547], "content": ", these are the values of ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [419, 534, 459, 546], "content": "a,b,c,d", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [459, 533, 540, 547], "content": " for which (3.5)", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [69, 547, 163, 562], "content": "is invertible. For ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [164, 549, 171, 558], "content": "k", "score": 0.87, "index": 84}, {"type": "text", "coordinates": [171, 547, 542, 562], "content": " odd, the group of simple-current automorphisms is isomorphic to the", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [70, 562, 163, 576], "content": "symmetric group ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [163, 564, 178, 574], "content": "\\mathfrak{S}_{3}", "score": 0.91, "index": 87}, {"type": "text", "coordinates": [179, 562, 264, 576], "content": " when 4 divides ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [264, 567, 270, 573], "content": "r", "score": 0.85, "index": 89}, {"type": "text", "coordinates": [271, 562, 314, 576], "content": ", and to ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [315, 563, 328, 574], "content": "\\mathbb{Z}_{6}", "score": 0.89, "index": 91}, {"type": "text", "coordinates": [329, 562, 363, 576], "content": " when ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [363, 564, 392, 573], "content": "r\\equiv2", "score": 0.85, "index": 93}, {"type": "text", "coordinates": [392, 562, 442, 576], "content": " (mod 4).", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [93, 575, 115, 592], "content": "For", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [116, 578, 145, 587], "content": "k=2", "score": 0.9, "index": 96}, {"type": "text", "coordinates": [145, 575, 167, 592], "content": " (so ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [167, 578, 201, 587], "content": "\\kappa=2r", "score": 0.86, "index": 98}, {"type": "text", "coordinates": [202, 575, 541, 592], "content": "), there are several Galois fusion-symmetries. In particular, write", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [71, 591, 160, 602], "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "score": 0.9, "index": 100}, {"type": "text", "coordinates": [160, 587, 183, 609], "content": " for ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [184, 593, 259, 603], "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "score": 0.91, "index": 102}, {"type": "text", "coordinates": [259, 587, 290, 609], "content": ", and ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [291, 591, 390, 604], "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "score": 0.92, "index": 104}, {"type": "text", "coordinates": [390, 587, 447, 609], "content": ". As with ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [447, 592, 469, 605], "content": "B_{r,2}", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [469, 587, 476, 609], "content": ", ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [477, 590, 527, 605], "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [511, 588, 540, 604], "content": "1 is", "score": 1.0, "index": 109}, {"type": "text", "coordinates": [71, 605, 169, 620], "content": "rational so for any ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [169, 610, 180, 615], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "score": 0.86, "index": 111}, {"type": "text", "coordinates": [180, 605, 241, 620], "content": " coprime to ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [241, 607, 253, 616], "content": "2r", "score": 0.86, "index": 113}, {"type": "text", "coordinates": [254, 605, 432, 620], "content": ", we get a Galois fusion-symmetry ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [432, 606, 462, 619], "content": "\\pi\\{m\\}", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [462, 605, 512, 620], "content": ". It takes ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [513, 607, 525, 616], "content": "\\lambda^{a}", "score": 0.9, "index": 117}, {"type": "text", "coordinates": [525, 605, 542, 620], "content": " to", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [71, 621, 93, 630], "content": "\\lambda^{m a}", "score": 0.89, "index": 119}, {"type": "text", "coordinates": [93, 617, 286, 635], "content": ", where the superscript is taken mod ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [286, 620, 299, 630], "content": "2r", "score": 0.8, "index": 121}, {"type": "text", "coordinates": [299, 617, 367, 635], "content": ", and will fix ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [367, 619, 386, 631], "content": "J_{v}0", "score": 0.88, "index": 123}, {"type": "text", "coordinates": [387, 617, 423, 635], "content": ". Also,", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [424, 620, 454, 632], "content": "\\pi\\{m\\}", "score": 0.94, "index": 125}, {"type": "text", "coordinates": [455, 617, 507, 635], "content": " will send ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [507, 621, 525, 631], "content": "J_{s}0", "score": 0.92, "index": 127}, {"type": "text", "coordinates": [526, 617, 542, 635], "content": " to", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [71, 635, 95, 647], "content": "J_{s}^{m}0", "score": 0.92, "index": 129}, {"type": "text", "coordinates": [95, 633, 241, 650], "content": ", as well as stabilise the set ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [241, 634, 372, 647], "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "score": 0.91, "index": 131}, {"type": "text", "coordinates": [373, 633, 481, 650], "content": ". (In particular, put ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [482, 636, 508, 644], "content": "t=r", "score": 0.9, "index": 133}, {"type": "text", "coordinates": [508, 633, 541, 650], "content": " when", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [71, 653, 77, 658], "content": "r", "score": 0.87, "index": 135}, {"type": "text", "coordinates": [77, 648, 164, 663], "content": " is even or when ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [165, 650, 198, 659], "content": "m\\equiv1", "score": 0.88, "index": 137}, {"type": "text", "coordinates": [198, 648, 324, 663], "content": " (mod 4), otherwise put ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [325, 649, 371, 660], "content": "t=r-1", "score": 0.87, "index": 139}, {"type": "text", "coordinates": [371, 648, 445, 663], "content": "; then for any ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [445, 646, 540, 662], "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "score": 0.84, "index": 141}, {"type": "text", "coordinates": [68, 661, 83, 681], "content": "is ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [83, 663, 123, 677], "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "score": 0.94, "index": 143}, {"type": "text", "coordinates": [123, 661, 141, 681], "content": " or ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [141, 663, 192, 677], "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [193, 661, 329, 681], "content": ", when the Jacobi symbol ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [329, 663, 350, 678], "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "score": 0.88, "index": 147}, {"type": "text", "coordinates": [351, 661, 365, 681], "content": " is ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [365, 664, 382, 676], "content": "\\pm1", "score": 0.82, "index": 149}, {"type": "text", "coordinates": [382, 661, 459, 681], "content": ", respectively.)", "score": 1.0, "index": 150}, {"type": "text", "coordinates": [93, 684, 323, 705], "content": "Theorem 3.D. The fusion-symmetries of ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [324, 685, 347, 700], "content": "D_{r}^{(1)}", "score": 0.93, "index": 152}, {"type": "text", "coordinates": [348, 684, 371, 705], "content": "for ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [371, 687, 405, 701], "content": "k\\neq2", "score": 0.89, "index": 154}, {"type": "text", "coordinates": [405, 684, 513, 705], "content": " are all of the form ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [513, 688, 536, 701], "content": "C_{i}\\,\\pi", "score": 0.88, "index": 156}, {"type": "text", "coordinates": [537, 684, 541, 705], "content": ",", "score": 1.0, "index": 157}, {"type": "text", "coordinates": [72, 702, 105, 718], "content": "where ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [105, 704, 118, 715], "content": "C_{i}", "score": 0.9, "index": 159}, {"type": "text", "coordinates": [118, 702, 272, 718], "content": " is a conjugation, and where ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [273, 708, 280, 713], "content": "\\pi", "score": 0.79, "index": 161}, {"type": "text", "coordinates": [281, 702, 541, 718], "content": " is a simple-current automorphism. Similarly for", "score": 1.0, "index": 162}]	[]	[{"type": "block", "coordinates": [205, 99, 405, 137], "content": "", "caption": ""}, {"type": "block", "coordinates": [195, 305, 415, 337], "content": "", "caption": ""}, {"type": "block", "coordinates": [164, 375, 448, 406], "content": "", "caption": ""}, {"type": "block", "coordinates": [126, 441, 490, 510], "content": "", "caption": ""}, {"type": "inline", "coordinates": [135, 75, 204, 86], "content": "1\\leq i<r-2", "caption": ""}, {"type": "inline", "coordinates": [230, 75, 259, 85], "content": "k>2", "caption": ""}, {"type": "inline", "coordinates": [106, 150, 209, 163], "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "caption": ""}, {"type": "inline", "coordinates": [396, 151, 418, 163], "content": "\\lambda(\\ell)", "caption": ""}, {"type": "inline", "coordinates": [503, 150, 541, 164], "content": "\\lambda(\\ell)\\,=", "caption": ""}, {"type": "inline", "coordinates": [71, 163, 171, 180], "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [482, 185, 487, 191], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [513, 182, 520, 191], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [174, 199, 180, 205], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [451, 196, 463, 207], "content": "J_{s}", "caption": ""}, {"type": "inline", "coordinates": [71, 210, 78, 219], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [405, 209, 521, 222], "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "caption": ""}, {"type": "inline", "coordinates": [71, 223, 122, 237], "content": "a\\in\\{0,2\\}", "caption": ""}, {"type": "inline", "coordinates": [182, 224, 190, 234], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [496, 224, 540, 236], "content": "\\pi=\\pi[a]", "caption": ""}, {"type": "inline", "coordinates": [95, 238, 172, 251], "content": "\\pi^{\\prime}=\\pi[a k-a]", "caption": ""}, {"type": "inline", "coordinates": [195, 238, 248, 250], "content": "0\\leq a\\leq3", "caption": ""}, {"type": "inline", "coordinates": [294, 239, 324, 248], "content": "k\\equiv2", "caption": ""}, {"type": "inline", "coordinates": [494, 239, 536, 250], "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "caption": ""}, {"type": "inline", "coordinates": [155, 253, 172, 265], "content": "4\|k", "caption": ""}, {"type": "inline", "coordinates": [242, 254, 255, 264], "content": "\\mathbb{Z}_{4}", "caption": ""}, {"type": "inline", "coordinates": [130, 271, 136, 277], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [408, 268, 421, 279], "content": "J_{v}", "caption": ""}, {"type": "inline", "coordinates": [448, 268, 460, 279], "content": "J_{s}", "caption": ""}, {"type": "inline", "coordinates": [71, 282, 78, 291], "content": "k", "caption": ""}, {"type": "inline", 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"content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "caption": ""}, {"type": "inline", "coordinates": [141, 663, 192, 677], "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "caption": ""}, {"type": "inline", "coordinates": [329, 663, 350, 678], "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "caption": ""}, {"type": "inline", "coordinates": [365, 664, 382, 676], "content": "\\pm1", "caption": ""}, {"type": "inline", "coordinates": [324, 685, 347, 700], "content": "D_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [371, 687, 405, 701], "content": "k\\neq2", "caption": ""}, {"type": "inline", "coordinates": [513, 688, 536, 701], "content": "C_{i}\\,\\pi", "caption": ""}, {"type": "inline", "coordinates": [105, 704, 118, 715], "content": "C_{i}", "caption": ""}, {"type": "inline", "coordinates": [273, 708, 280, 713], "content": "\\pi", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "valid for all $1\\leq i<r-2$ and $k>2$ . We also will use the character formula (2.1b) ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)$ and the orthonormal components $\\lambda(\\ell)$ are defined by $\\lambda(\\ell)\\,=$ $\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}$ . ", "page_idx": 11}, {"type": "text", "text": "The simple-current automorphisms are as follows, and depend on whether $r$ and $k$ are even or odd. When $r$ is odd, the group of simple-currents is generated by $J_{s}$ . If in addition $k$ is odd, there will be only two simple-current automorphisms: $\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}$ for $a\\in\\{0,2\\}$ . If instead $k$ is even, there will be four simple-current automorphisms: $\\pi=\\pi[a]$ and $\\pi^{\\prime}=\\pi[a k-a]$ for $0\\leq a\\leq3$ . When $k\\equiv2$ (mod 4), these form the group $\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$ , otherwise when $4\|k$ the group is $\\mathbb{Z}_{4}$ . ", "page_idx": 11}, {"type": "text", "text": "When $r$ is even, the simple-currents are generated by both $J_{v}$ and $J_{s}$ . If in addition $k$ is even, we have 16 simple-current automorphisms: ", "page_idx": 11}, {"type": "equation", "text": "$$\n{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "for any $a,b,c,d\\in\\{0,1\\}$ , forming a group isomorphic to $\\mathbb{Z}_{2}^{4}$ . This notation means ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "When $k$ is odd, we will have six simple-current automorphisms: ", "page_idx": 11}, {"type": "equation", "text": "$$\n{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "where $a={\\frac{r}{2}}$ or $d=0$ , and where $b=1$ or $d=1$ . The corresponding permutation of $P_{+}$ is still given by (3.5). Again, for these $r,k$ , these are the values of $a,b,c,d$ for which (3.5) is invertible. For $k$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $\\mathfrak{S}_{3}$ when 4 divides $r$ , and to $\\mathbb{Z}_{6}$ when $r\\equiv2$ (mod 4). ", "page_idx": 11}, {"type": "text", "text": "For $k=2$ (so $\\kappa=2r$ ), there are several Galois fusion-symmetries. In particular, write $\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}$ for $1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2$ , and $\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}$ . As with $B_{r,2}$ , $\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}$ 1 is rational so for any ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ coprime to $2r$ , we get a Galois fusion-symmetry $\\pi\\{m\\}$ . It takes $\\lambda^{a}$ to $\\lambda^{m a}$ , where the superscript is taken mod $2r$ , and will fix $J_{v}0$ . Also, $\\pi\\{m\\}$ will send $J_{s}0$ to $J_{s}^{m}0$ , as well as stabilise the set $\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}$ . (In particular, put $t=r$ when $r$ is even or when $m\\equiv1$ (mod 4), otherwise put $t=r-1$ ; then for any $i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}$ is $C_{1}^{j}J_{v}^{i}\\Lambda_{t}$ or $C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}$ , when the Jacobi symbol $\\textstyle\\left({\\frac{\\kappa}{m}}\\right)$ is $\\pm1$ , respectively.) ", "page_idx": 11}, {"type": "text", "text": "Theorem 3.D. The fusion-symmetries of $D_{r}^{(1)}$ for $k\\neq2$ are all of the form $C_{i}\\,\\pi$ , where $C_{i}$ is a conjugation, and where $\\pi$ is a simple-current automorphism. Similarly for ${D}_{4}^{(1)}$ at $k=2$ . Finally, when both $k=2$ and $r>4$ , any fusion-symmetry $\\pi$ can be written as $\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}$ for $a,b\\in\\{0,1\\}$ and any $m\\in\\mathbb{Z}_{2r}^{\\times}$ , $1\\leq m<r$ . 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We also will use the character formula (2.1b)", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 147, 541, 178], "lines": [{"bbox": [70, 149, 541, 166], "spans": [{"bbox": [70, 149, 106, 166], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 150, 209, 163], "score": 0.92, "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [209, 149, 396, 166], "score": 1.0, "content": " and the orthonormal components ", "type": "text"}, {"bbox": [396, 151, 418, 163], "score": 0.93, "content": "\\lambda(\\ell)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [418, 149, 503, 166], "score": 1.0, "content": " are defined by ", "type": "text"}, {"bbox": [503, 150, 541, 164], "score": 0.91, "content": "\\lambda(\\ell)\\,=", "type": "inline_equation", "height": 14, "width": 38}], "index": 2}, {"bbox": [71, 162, 175, 180], "spans": [{"bbox": [71, 163, 171, 180], "score": 0.93, "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "type": "inline_equation", "height": 17, "width": 100}, {"bbox": [171, 162, 175, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [70, 178, 542, 264], "lines": [{"bbox": [76, 178, 542, 196], "spans": [{"bbox": [76, 178, 481, 196], "score": 1.0, "content": "The simple-current automorphisms are as follows, and depend on whether ", "type": "text"}, {"bbox": [482, 185, 487, 191], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [488, 178, 513, 196], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [513, 182, 520, 191], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [520, 178, 542, 196], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [69, 194, 541, 209], "spans": [{"bbox": [69, 194, 173, 209], "score": 1.0, "content": "even or odd. When", "type": "text"}, {"bbox": [174, 199, 180, 205], "score": 0.67, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [180, 194, 451, 209], "score": 1.0, "content": " is odd, the group of simple-currents is generated by ", "type": "text"}, {"bbox": [451, 196, 463, 207], "score": 0.9, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [464, 194, 541, 209], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 5}, {"bbox": [71, 207, 543, 224], "spans": [{"bbox": [71, 210, 78, 219], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 207, 404, 224], "score": 1.0, "content": " is odd, there will be only two simple-current automorphisms: ", "type": "text"}, {"bbox": [405, 209, 521, 222], "score": 0.93, "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [521, 207, 543, 224], "score": 1.0, "content": "for", "type": "text"}], "index": 6}, {"bbox": [71, 222, 540, 239], "spans": [{"bbox": [71, 223, 122, 237], "score": 0.92, "content": "a\\in\\{0,2\\}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [122, 222, 182, 239], "score": 1.0, "content": ". If instead ", "type": "text"}, {"bbox": [182, 224, 190, 234], "score": 0.84, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 222, 495, 239], "score": 1.0, "content": " is even, there will be four simple-current automorphisms: ", "type": "text"}, {"bbox": [496, 224, 540, 236], "score": 0.93, "content": "\\pi=\\pi[a]", "type": "inline_equation", "height": 12, "width": 44}], "index": 7}, {"bbox": [70, 236, 541, 254], "spans": [{"bbox": [70, 236, 95, 254], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 238, 172, 251], "score": 0.93, "content": "\\pi^{\\prime}=\\pi[a k-a]", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [172, 236, 194, 254], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [195, 238, 248, 250], "score": 0.85, "content": "0\\leq a\\leq3", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [249, 236, 293, 254], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [294, 239, 324, 248], "score": 0.92, "content": "k\\equiv2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [324, 236, 493, 254], "score": 1.0, "content": " (mod 4), these form the group ", "type": "text"}, {"bbox": [494, 239, 536, 250], "score": 0.92, "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [536, 236, 541, 254], "score": 1.0, "content": ",", "type": "text"}], "index": 8}, {"bbox": [70, 250, 260, 268], "spans": [{"bbox": [70, 250, 155, 268], "score": 1.0, "content": "otherwise when ", "type": "text"}, {"bbox": [155, 253, 172, 265], "score": 0.88, "content": "4\|k", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [172, 250, 241, 268], "score": 1.0, "content": " the group is ", "type": "text"}, {"bbox": [242, 254, 255, 264], "score": 0.9, "content": "\\mathbb{Z}_{4}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [256, 250, 260, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 6.5}, {"type": "text", "bbox": [70, 264, 541, 293], "lines": [{"bbox": [95, 266, 541, 281], "spans": [{"bbox": [95, 266, 130, 281], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 271, 136, 277], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [136, 266, 408, 281], "score": 1.0, "content": " is even, the simple-currents are generated by both ", "type": "text"}, {"bbox": [408, 268, 421, 279], "score": 0.92, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [421, 266, 448, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [448, 268, 460, 279], "score": 0.91, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [461, 266, 541, 281], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 10}, {"bbox": [71, 280, 349, 297], "spans": [{"bbox": [71, 282, 78, 291], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 280, 349, 297], "score": 1.0, "content": " is even, we have 16 simple-current automorphisms:", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 346, 495, 362], "lines": [{"bbox": [70, 348, 497, 366], "spans": [{"bbox": [70, 348, 112, 366], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [112, 351, 195, 363], "score": 0.93, "content": "a,b,c,d\\in\\{0,1\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [195, 348, 365, 366], "score": 1.0, "content": ", forming a group isomorphic to ", "type": "text"}, {"bbox": [365, 350, 379, 363], "score": 0.92, "content": "\\mathbb{Z}_{2}^{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [379, 348, 497, 366], "score": 1.0, "content": ". This notation means", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 415, 407, 430], "lines": [{"bbox": [71, 417, 405, 432], "spans": [{"bbox": [71, 417, 106, 432], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 420, 113, 429], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [113, 417, 405, 432], "score": 1.0, "content": " is odd, we will have six simple-current automorphisms:", "type": "text"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [69, 516, 541, 574], "lines": [{"bbox": [70, 517, 539, 536], "spans": [{"bbox": [70, 517, 106, 536], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 520, 137, 533], "score": 0.93, "content": "a={\\frac{r}{2}}", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [138, 517, 156, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [156, 520, 186, 529], "score": 0.89, "content": "d=0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [187, 517, 252, 536], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [252, 519, 281, 529], "score": 0.88, "content": "b=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 517, 299, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [299, 520, 329, 529], "score": 0.89, "content": "d=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [330, 517, 523, 536], "score": 1.0, "content": ". The corresponding permutation of ", "type": "text"}, {"bbox": [523, 520, 539, 532], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}], "index": 17}, {"bbox": [70, 533, 540, 547], "spans": [{"bbox": [70, 533, 274, 547], "score": 1.0, "content": "is still given by (3.5). Again, for these ", "type": "text"}, {"bbox": [275, 533, 293, 546], "score": 0.87, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [293, 533, 419, 547], "score": 1.0, "content": ", these are the values of ", "type": "text"}, {"bbox": [419, 534, 459, 546], "score": 0.91, "content": "a,b,c,d", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [459, 533, 540, 547], "score": 1.0, "content": " for which (3.5)", "type": "text"}], "index": 18}, {"bbox": [69, 547, 542, 562], "spans": [{"bbox": [69, 547, 163, 562], "score": 1.0, "content": "is invertible. For ", "type": "text"}, {"bbox": [164, 549, 171, 558], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [171, 547, 542, 562], "score": 1.0, "content": " odd, the group of simple-current automorphisms is isomorphic to the", "type": "text"}], "index": 19}, {"bbox": [70, 562, 442, 576], "spans": [{"bbox": [70, 562, 163, 576], "score": 1.0, "content": "symmetric group ", "type": "text"}, {"bbox": [163, 564, 178, 574], "score": 0.91, "content": "\\mathfrak{S}_{3}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [179, 562, 264, 576], "score": 1.0, "content": " when 4 divides ", "type": "text"}, {"bbox": [264, 567, 270, 573], "score": 0.85, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [271, 562, 314, 576], "score": 1.0, "content": ", and to ", "type": "text"}, {"bbox": [315, 563, 328, 574], "score": 0.89, "content": "\\mathbb{Z}_{6}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [329, 562, 363, 576], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [363, 564, 392, 573], "score": 0.85, "content": "r\\equiv2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [392, 562, 442, 576], "score": 1.0, "content": " (mod 4).", "type": "text"}], "index": 20}], "index": 18.5}, {"type": "text", "bbox": [70, 574, 542, 678], "lines": [{"bbox": [93, 575, 541, 592], "spans": [{"bbox": [93, 575, 115, 592], "score": 1.0, "content": "For", "type": "text"}, {"bbox": [116, 578, 145, 587], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [145, 575, 167, 592], "score": 1.0, "content": " (so ", "type": "text"}, {"bbox": [167, 578, 201, 587], "score": 0.86, "content": "\\kappa=2r", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [202, 575, 541, 592], "score": 1.0, "content": "), there are several Galois fusion-symmetries. In particular, write", "type": "text"}], "index": 21}, {"bbox": [71, 587, 540, 609], "spans": [{"bbox": [71, 591, 160, 602], "score": 0.9, "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [160, 587, 183, 609], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [184, 593, 259, 603], "score": 0.91, "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [259, 587, 290, 609], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [291, 591, 390, 604], "score": 0.92, "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [390, 587, 447, 609], "score": 1.0, "content": ". As with ", "type": "text"}, {"bbox": [447, 592, 469, 605], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [469, 587, 476, 609], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 590, 527, 605], "score": 0.93, "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [511, 588, 540, 604], "score": 1.0, "content": "1 is", "type": "text"}], "index": 22}, {"bbox": [71, 605, 542, 620], "spans": [{"bbox": [71, 605, 169, 620], "score": 1.0, "content": "rational so for any ", "type": "text"}, {"bbox": [169, 610, 180, 615], "score": 0.86, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [180, 605, 241, 620], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [241, 607, 253, 616], "score": 0.86, "content": "2r", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [254, 605, 432, 620], "score": 1.0, "content": ", we get a Galois fusion-symmetry ", "type": "text"}, {"bbox": [432, 606, 462, 619], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [462, 605, 512, 620], "score": 1.0, "content": ". It takes ", "type": "text"}, {"bbox": [513, 607, 525, 616], "score": 0.9, "content": "\\lambda^{a}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [525, 605, 542, 620], "score": 1.0, "content": " to", "type": "text"}], "index": 23}, {"bbox": [71, 617, 542, 635], "spans": [{"bbox": [71, 621, 93, 630], "score": 0.89, "content": "\\lambda^{m a}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [93, 617, 286, 635], "score": 1.0, "content": ", where the superscript is taken mod ", "type": "text"}, {"bbox": [286, 620, 299, 630], "score": 0.8, "content": "2r", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [299, 617, 367, 635], "score": 1.0, "content": ", and will fix ", "type": "text"}, {"bbox": [367, 619, 386, 631], "score": 0.88, "content": "J_{v}0", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [387, 617, 423, 635], "score": 1.0, "content": ". Also,", "type": "text"}, {"bbox": [424, 620, 454, 632], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [455, 617, 507, 635], "score": 1.0, "content": " will send ", "type": "text"}, {"bbox": [507, 621, 525, 631], "score": 0.92, "content": "J_{s}0", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [526, 617, 542, 635], "score": 1.0, "content": " to", "type": "text"}], "index": 24}, {"bbox": [71, 633, 541, 650], "spans": [{"bbox": [71, 635, 95, 647], "score": 0.92, "content": "J_{s}^{m}0", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [95, 633, 241, 650], "score": 1.0, "content": ", as well as stabilise the set ", "type": "text"}, {"bbox": [241, 634, 372, 647], "score": 0.91, "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "type": "inline_equation", "height": 13, "width": 131}, {"bbox": [373, 633, 481, 650], "score": 1.0, "content": ". (In particular, put ", "type": "text"}, {"bbox": [482, 636, 508, 644], "score": 0.9, "content": "t=r", "type": "inline_equation", "height": 8, "width": 26}, {"bbox": [508, 633, 541, 650], "score": 1.0, "content": " when", "type": "text"}], "index": 25}, {"bbox": [71, 646, 540, 663], "spans": [{"bbox": [71, 653, 77, 658], "score": 0.87, "content": "r", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [77, 648, 164, 663], "score": 1.0, "content": " is even or when ", "type": "text"}, {"bbox": [165, 650, 198, 659], "score": 0.88, "content": "m\\equiv1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [198, 648, 324, 663], "score": 1.0, "content": " (mod 4), otherwise put ", "type": "text"}, {"bbox": [325, 649, 371, 660], "score": 0.87, "content": "t=r-1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [371, 648, 445, 663], "score": 1.0, "content": "; then for any ", "type": "text"}, {"bbox": [445, 646, 540, 662], "score": 0.84, "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "type": "inline_equation", "height": 16, "width": 95}], "index": 26}, {"bbox": [68, 661, 459, 681], "spans": [{"bbox": [68, 661, 83, 681], "score": 1.0, "content": "is ", "type": "text"}, {"bbox": [83, 663, 123, 677], "score": 0.94, "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [123, 661, 141, 681], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [141, 663, 192, 677], "score": 0.93, "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [193, 661, 329, 681], "score": 1.0, "content": ", when the Jacobi symbol ", "type": "text"}, {"bbox": [329, 663, 350, 678], "score": 0.88, "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [351, 661, 365, 681], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [365, 664, 382, 676], "score": 0.82, "content": "\\pm1", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [382, 661, 459, 681], "score": 1.0, "content": ", respectively.)", "type": "text"}], "index": 27}], "index": 24}, {"type": "text", "bbox": [70, 684, 541, 716], "lines": [{"bbox": [93, 684, 541, 705], "spans": [{"bbox": [93, 684, 323, 705], "score": 1.0, "content": "Theorem 3.D. The fusion-symmetries of ", "type": "text"}, {"bbox": [324, 685, 347, 700], "score": 0.93, "content": "D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [348, 684, 371, 705], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [371, 687, 405, 701], "score": 0.89, "content": "k\\neq2", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [405, 684, 513, 705], "score": 1.0, "content": " are all of the form ", "type": "text"}, {"bbox": [513, 688, 536, 701], "score": 0.88, "content": "C_{i}\\,\\pi", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [537, 684, 541, 705], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [72, 702, 541, 718], "spans": [{"bbox": [72, 702, 105, 718], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 704, 118, 715], "score": 0.9, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [118, 702, 272, 718], "score": 1.0, "content": " is a conjugation, and where ", "type": "text"}, {"bbox": [273, 708, 280, 713], "score": 0.79, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [281, 702, 541, 718], "score": 1.0, "content": " is a simple-current automorphism. Similarly for", "type": "text"}], "index": 29}], "index": 28.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 505, 86], "lines": [{"bbox": [71, 73, 504, 88], "spans": [{"bbox": [71, 73, 135, 88], "score": 1.0, "content": "valid for all ", "type": "text"}, {"bbox": [135, 75, 204, 86], "score": 0.88, "content": "1\\leq i<r-2", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [204, 73, 230, 88], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [230, 75, 259, 85], "score": 0.88, "content": "k>2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [259, 73, 504, 88], "score": 1.0, "content": ". We also will use the character formula (2.1b)", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [71, 73, 504, 88]}, {"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 147, 541, 178], "lines": [{"bbox": [70, 149, 541, 166], "spans": [{"bbox": [70, 149, 106, 166], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 150, 209, 163], "score": 0.92, "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [209, 149, 396, 166], "score": 1.0, "content": " and the orthonormal components ", "type": "text"}, {"bbox": [396, 151, 418, 163], "score": 0.93, "content": "\\lambda(\\ell)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [418, 149, 503, 166], "score": 1.0, "content": " are defined by ", "type": "text"}, {"bbox": [503, 150, 541, 164], "score": 0.91, "content": "\\lambda(\\ell)\\,=", "type": "inline_equation", "height": 14, "width": 38}], "index": 2}, {"bbox": [71, 162, 175, 180], "spans": [{"bbox": [71, 163, 171, 180], "score": 0.93, "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "type": "inline_equation", "height": 17, "width": 100}, {"bbox": [171, 162, 175, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 149, 541, 180]}, {"type": "text", "bbox": [70, 178, 542, 264], "lines": [{"bbox": [76, 178, 542, 196], "spans": [{"bbox": [76, 178, 481, 196], "score": 1.0, "content": "The simple-current automorphisms are as follows, and depend on whether ", "type": "text"}, {"bbox": [482, 185, 487, 191], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [488, 178, 513, 196], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [513, 182, 520, 191], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [520, 178, 542, 196], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [69, 194, 541, 209], "spans": [{"bbox": [69, 194, 173, 209], "score": 1.0, "content": "even or odd. When", "type": "text"}, {"bbox": [174, 199, 180, 205], "score": 0.67, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [180, 194, 451, 209], "score": 1.0, "content": " is odd, the group of simple-currents is generated by ", "type": "text"}, {"bbox": [451, 196, 463, 207], "score": 0.9, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [464, 194, 541, 209], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 5}, {"bbox": [71, 207, 543, 224], "spans": [{"bbox": [71, 210, 78, 219], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 207, 404, 224], "score": 1.0, "content": " is odd, there will be only two simple-current automorphisms: ", "type": "text"}, {"bbox": [405, 209, 521, 222], "score": 0.93, "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [521, 207, 543, 224], "score": 1.0, "content": "for", "type": "text"}], "index": 6}, {"bbox": [71, 222, 540, 239], "spans": [{"bbox": [71, 223, 122, 237], "score": 0.92, "content": "a\\in\\{0,2\\}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [122, 222, 182, 239], "score": 1.0, "content": ". If instead ", "type": "text"}, {"bbox": [182, 224, 190, 234], "score": 0.84, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 222, 495, 239], "score": 1.0, "content": " is even, there will be four simple-current automorphisms: ", "type": "text"}, {"bbox": [496, 224, 540, 236], "score": 0.93, "content": "\\pi=\\pi[a]", "type": "inline_equation", "height": 12, "width": 44}], "index": 7}, {"bbox": [70, 236, 541, 254], "spans": [{"bbox": [70, 236, 95, 254], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 238, 172, 251], "score": 0.93, "content": "\\pi^{\\prime}=\\pi[a k-a]", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [172, 236, 194, 254], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [195, 238, 248, 250], "score": 0.85, "content": "0\\leq a\\leq3", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [249, 236, 293, 254], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [294, 239, 324, 248], "score": 0.92, "content": "k\\equiv2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [324, 236, 493, 254], "score": 1.0, "content": " (mod 4), these form the group ", "type": "text"}, {"bbox": [494, 239, 536, 250], "score": 0.92, "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [536, 236, 541, 254], "score": 1.0, "content": ",", "type": "text"}], "index": 8}, {"bbox": [70, 250, 260, 268], "spans": [{"bbox": [70, 250, 155, 268], "score": 1.0, "content": "otherwise when ", "type": "text"}, {"bbox": [155, 253, 172, 265], "score": 0.88, "content": "4\|k", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [172, 250, 241, 268], "score": 1.0, "content": " the group is ", "type": "text"}, {"bbox": [242, 254, 255, 264], "score": 0.9, "content": "\\mathbb{Z}_{4}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [256, 250, 260, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 6.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [69, 178, 543, 268]}, {"type": "text", "bbox": [70, 264, 541, 293], "lines": [{"bbox": [95, 266, 541, 281], "spans": [{"bbox": [95, 266, 130, 281], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 271, 136, 277], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [136, 266, 408, 281], "score": 1.0, "content": " is even, the simple-currents are generated by both ", "type": "text"}, {"bbox": [408, 268, 421, 279], "score": 0.92, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [421, 266, 448, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [448, 268, 460, 279], "score": 0.91, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [461, 266, 541, 281], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 10}, {"bbox": [71, 280, 349, 297], "spans": [{"bbox": [71, 282, 78, 291], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 280, 349, 297], "score": 1.0, "content": " is even, we have 16 simple-current automorphisms:", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [71, 266, 541, 297]}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 346, 495, 362], "lines": [{"bbox": [70, 348, 497, 366], "spans": [{"bbox": [70, 348, 112, 366], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [112, 351, 195, 363], "score": 0.93, "content": "a,b,c,d\\in\\{0,1\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [195, 348, 365, 366], "score": 1.0, "content": ", forming a group isomorphic to ", "type": "text"}, {"bbox": [365, 350, 379, 363], "score": 0.92, "content": "\\mathbb{Z}_{2}^{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [379, 348, 497, 366], "score": 1.0, "content": ". This notation means", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 348, 497, 366]}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 415, 407, 430], "lines": [{"bbox": [71, 417, 405, 432], "spans": [{"bbox": [71, 417, 106, 432], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 420, 113, 429], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [113, 417, 405, 432], "score": 1.0, "content": " is odd, we will have six simple-current automorphisms:", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [71, 417, 405, 432]}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 516, 541, 574], "lines": [{"bbox": [70, 517, 539, 536], "spans": [{"bbox": [70, 517, 106, 536], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 520, 137, 533], "score": 0.93, "content": "a={\\frac{r}{2}}", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [138, 517, 156, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [156, 520, 186, 529], "score": 0.89, "content": "d=0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [187, 517, 252, 536], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [252, 519, 281, 529], "score": 0.88, "content": "b=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 517, 299, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [299, 520, 329, 529], "score": 0.89, "content": "d=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [330, 517, 523, 536], "score": 1.0, "content": ". The corresponding permutation of ", "type": "text"}, {"bbox": [523, 520, 539, 532], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}], "index": 17}, {"bbox": [70, 533, 540, 547], "spans": [{"bbox": [70, 533, 274, 547], "score": 1.0, "content": "is still given by (3.5). Again, for these ", "type": "text"}, {"bbox": [275, 533, 293, 546], "score": 0.87, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [293, 533, 419, 547], "score": 1.0, "content": ", these are the values of ", "type": "text"}, {"bbox": [419, 534, 459, 546], "score": 0.91, "content": "a,b,c,d", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [459, 533, 540, 547], "score": 1.0, "content": " for which (3.5)", "type": "text"}], "index": 18}, {"bbox": [69, 547, 542, 562], "spans": [{"bbox": [69, 547, 163, 562], "score": 1.0, "content": "is invertible. For ", "type": "text"}, {"bbox": [164, 549, 171, 558], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [171, 547, 542, 562], "score": 1.0, "content": " odd, the group of simple-current automorphisms is isomorphic to the", "type": "text"}], "index": 19}, {"bbox": [70, 562, 442, 576], "spans": [{"bbox": [70, 562, 163, 576], "score": 1.0, "content": "symmetric group ", "type": "text"}, {"bbox": [163, 564, 178, 574], "score": 0.91, "content": "\\mathfrak{S}_{3}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [179, 562, 264, 576], "score": 1.0, "content": " when 4 divides ", "type": "text"}, {"bbox": [264, 567, 270, 573], "score": 0.85, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [271, 562, 314, 576], "score": 1.0, "content": ", and to ", "type": "text"}, {"bbox": [315, 563, 328, 574], "score": 0.89, "content": "\\mathbb{Z}_{6}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [329, 562, 363, 576], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [363, 564, 392, 573], "score": 0.85, "content": "r\\equiv2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [392, 562, 442, 576], "score": 1.0, "content": " (mod 4).", "type": "text"}], "index": 20}], "index": 18.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [69, 517, 542, 576]}, {"type": "text", "bbox": [70, 574, 542, 678], "lines": [{"bbox": [93, 575, 541, 592], "spans": [{"bbox": [93, 575, 115, 592], "score": 1.0, "content": "For", "type": "text"}, {"bbox": [116, 578, 145, 587], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [145, 575, 167, 592], "score": 1.0, "content": " (so ", "type": "text"}, {"bbox": [167, 578, 201, 587], "score": 0.86, "content": "\\kappa=2r", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [202, 575, 541, 592], "score": 1.0, "content": "), there are several Galois fusion-symmetries. In particular, write", "type": "text"}], "index": 21}, {"bbox": [71, 587, 540, 609], "spans": [{"bbox": [71, 591, 160, 602], "score": 0.9, "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [160, 587, 183, 609], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [184, 593, 259, 603], "score": 0.91, "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [259, 587, 290, 609], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [291, 591, 390, 604], "score": 0.92, "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [390, 587, 447, 609], "score": 1.0, "content": ". As with ", "type": "text"}, {"bbox": [447, 592, 469, 605], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [469, 587, 476, 609], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 590, 527, 605], "score": 0.93, "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [511, 588, 540, 604], "score": 1.0, "content": "1 is", "type": "text"}], "index": 22}, {"bbox": [71, 605, 542, 620], "spans": [{"bbox": [71, 605, 169, 620], "score": 1.0, "content": "rational so for any ", "type": "text"}, {"bbox": [169, 610, 180, 615], "score": 0.86, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [180, 605, 241, 620], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [241, 607, 253, 616], "score": 0.86, "content": "2r", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [254, 605, 432, 620], "score": 1.0, "content": ", we get a Galois fusion-symmetry ", "type": "text"}, {"bbox": [432, 606, 462, 619], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [462, 605, 512, 620], "score": 1.0, "content": ". It takes ", "type": "text"}, {"bbox": [513, 607, 525, 616], "score": 0.9, "content": "\\lambda^{a}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [525, 605, 542, 620], "score": 1.0, "content": " to", "type": "text"}], "index": 23}, {"bbox": [71, 617, 542, 635], "spans": [{"bbox": [71, 621, 93, 630], "score": 0.89, "content": "\\lambda^{m a}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [93, 617, 286, 635], "score": 1.0, "content": ", where the superscript is taken mod ", "type": "text"}, {"bbox": [286, 620, 299, 630], "score": 0.8, "content": "2r", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [299, 617, 367, 635], "score": 1.0, "content": ", and will fix ", "type": "text"}, {"bbox": [367, 619, 386, 631], "score": 0.88, "content": "J_{v}0", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [387, 617, 423, 635], "score": 1.0, "content": ". Also,", "type": "text"}, {"bbox": [424, 620, 454, 632], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [455, 617, 507, 635], "score": 1.0, "content": " will send ", "type": "text"}, {"bbox": [507, 621, 525, 631], "score": 0.92, "content": "J_{s}0", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [526, 617, 542, 635], "score": 1.0, "content": " to", "type": "text"}], "index": 24}, {"bbox": [71, 633, 541, 650], "spans": [{"bbox": [71, 635, 95, 647], "score": 0.92, "content": "J_{s}^{m}0", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [95, 633, 241, 650], "score": 1.0, "content": ", as well as stabilise the set ", "type": "text"}, {"bbox": [241, 634, 372, 647], "score": 0.91, "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "type": "inline_equation", "height": 13, "width": 131}, {"bbox": [373, 633, 481, 650], "score": 1.0, "content": ". (In particular, put ", "type": "text"}, {"bbox": [482, 636, 508, 644], "score": 0.9, "content": "t=r", "type": "inline_equation", "height": 8, "width": 26}, {"bbox": [508, 633, 541, 650], "score": 1.0, "content": " when", "type": "text"}], "index": 25}, {"bbox": [71, 646, 540, 663], "spans": [{"bbox": [71, 653, 77, 658], "score": 0.87, "content": "r", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [77, 648, 164, 663], "score": 1.0, "content": " is even or when ", "type": "text"}, {"bbox": [165, 650, 198, 659], "score": 0.88, "content": "m\\equiv1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [198, 648, 324, 663], "score": 1.0, "content": " (mod 4), otherwise put ", "type": "text"}, {"bbox": [325, 649, 371, 660], "score": 0.87, "content": "t=r-1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [371, 648, 445, 663], "score": 1.0, "content": "; then for any ", "type": "text"}, {"bbox": [445, 646, 540, 662], "score": 0.84, "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "type": "inline_equation", "height": 16, "width": 95}], "index": 26}, {"bbox": [68, 661, 459, 681], "spans": [{"bbox": [68, 661, 83, 681], "score": 1.0, "content": "is ", "type": "text"}, {"bbox": [83, 663, 123, 677], "score": 0.94, "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [123, 661, 141, 681], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [141, 663, 192, 677], "score": 0.93, "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [193, 661, 329, 681], "score": 1.0, "content": ", when the Jacobi symbol ", "type": "text"}, {"bbox": [329, 663, 350, 678], "score": 0.88, "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [351, 661, 365, 681], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [365, 664, 382, 676], "score": 0.82, "content": "\\pm1", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [382, 661, 459, 681], "score": 1.0, "content": ", respectively.)", "type": "text"}], "index": 27}], "index": 24, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [68, 575, 542, 681]}, {"type": "text", "bbox": [70, 684, 541, 716], "lines": [{"bbox": [93, 684, 541, 705], "spans": [{"bbox": [93, 684, 323, 705], "score": 1.0, "content": "Theorem 3.D. The fusion-symmetries of ", "type": "text"}, {"bbox": [324, 685, 347, 700], "score": 0.93, "content": "D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [348, 684, 371, 705], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [371, 687, 405, 701], "score": 0.89, "content": "k\\neq2", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [405, 684, 513, 705], "score": 1.0, "content": " are all of the form ", "type": "text"}, {"bbox": [513, 688, 536, 701], "score": 0.88, "content": "C_{i}\\,\\pi", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [537, 684, 541, 705], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [72, 702, 541, 718], "spans": [{"bbox": [72, 702, 105, 718], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 704, 118, 715], "score": 0.9, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [118, 702, 272, 718], "score": 1.0, "content": " is a conjugation, and where ", "type": "text"}, {"bbox": [273, 708, 280, 713], "score": 0.79, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [281, 702, 541, 718], "score": 1.0, "content": " is a simple-current automorphism. Similarly for", "type": "text"}], "index": 29}, {"bbox": [71, 67, 544, 92], "spans": [{"bbox": [71, 70, 95, 88], "score": 0.9, "content": "{D}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24, "cross_page": true}, {"bbox": [95, 67, 112, 92], "score": 1.0, "content": "at ", "type": "text", "cross_page": true}, {"bbox": [112, 74, 142, 86], "score": 0.86, "content": "k=2", "type": "inline_equation", "height": 12, "width": 30, "cross_page": true}, {"bbox": [142, 67, 249, 92], "score": 1.0, "content": ". Finally, when both", "type": "text", "cross_page": true}, {"bbox": [250, 74, 280, 86], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 12, "width": 30, "cross_page": true}, {"bbox": [280, 67, 306, 92], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [306, 74, 335, 86], "score": 0.87, "content": "r>4", "type": "inline_equation", "height": 12, "width": 29, "cross_page": true}, {"bbox": [335, 67, 454, 92], "score": 1.0, "content": ", any fusion-symmetry ", "type": "text", "cross_page": true}, {"bbox": [454, 77, 463, 85], "score": 0.63, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [463, 67, 544, 92], "score": 1.0, "content": " can be written", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 86, 418, 106], "spans": [{"bbox": [70, 86, 86, 106], "score": 1.0, "content": "as ", "type": "text", "cross_page": true}, {"bbox": [86, 87, 171, 102], "score": 0.89, "content": "\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 85, "cross_page": true}, {"bbox": [172, 86, 194, 106], "score": 1.0, "content": " for ", "type": "text", "cross_page": true}, {"bbox": [194, 88, 255, 102], "score": 0.91, "content": "a,b\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 61, "cross_page": true}, {"bbox": [256, 86, 305, 106], "score": 1.0, "content": " and any ", "type": "text", "cross_page": true}, {"bbox": [305, 87, 349, 102], "score": 0.9, "content": "m\\in\\mathbb{Z}_{2r}^{\\times}", "type": "inline_equation", "height": 15, "width": 44, "cross_page": true}, {"bbox": [349, 86, 356, 106], "score": 1.0, "content": ", ", "type": "text", "cross_page": true}, {"bbox": [356, 88, 411, 102], "score": 0.9, "content": "1\\leq m<r", "type": "inline_equation", "height": 14, "width": 55, "cross_page": true}, {"bbox": [412, 86, 418, 106], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 1}], "index": 28.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [72, 684, 541, 718]}]}
0002044v1	7	$$\mathrm{ch}_{\overline{{{\mu}}}}$$ for $$X_{r}$$ can be written as a polynomial in the fundamental characters $$\mathrm{ch}_{\Lambda_{1}},\ldots,\mathrm{ch}_{\Lambda_{r}}$$ , we know from (2.1b) that $$\Gamma=\{\Lambda_{1},\dots,\Lambda_{r}\}$$ is a fusion-generator for $$X_{r}^{(1)}$$ at any level $$k$$ sufficiently large that $$P_{+}$$ contains all $$\Lambda_{i}$$ (in other words, for any $$k\geq\operatorname*{max}_{i}a_{i}^{\vee}$$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $$X_{r,k}$$ is $$\{\Lambda_{1},\ldots,\Lambda_{r}\}\cap{\cal P}_{+}$$ . Smaller fusion-generators usually exist — for example $$\{\Lambda_{1}\}$$ is a fusion-generator for $$A_{8,k}$$ whenever $$k$$ is even and coprime to 3. 2.3. Standard constructions of fusion-symmetries Simple-currents are a large source of fusion-symmetries. Let $$j$$ be any simple-current of order $${\boldsymbol{n}}$$ . Choose any number $$a\in\{0,1,\ldots,n-1\}$$ such that Any solution to this defines a fusion-symmetry $$\lambda\mapsto J^{n a Q_{j}\,(\lambda)}\lambda$$ , which we shall denote $$\pi[a]$$ or $$\pi_{j}[a]$$ . Note that from (2.2b), (2.5b) and (2.5c) that any $$\pi\,=\,\pi[a]$$ , $$a\in\mathbb{Z}$$ , obeys the relation $$N_{\pi\lambda,\pi\mu}^{\pi\nu}=\ N_{\lambda\mu}^{\nu}$$ when $$N_{\lambda\mu}^{\nu}\neq0$$ (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces $$\pi[a]$$ to be a permutation. Choosing $$b\equiv-a\,(n a Q_{j}(j)\!+\!1)^{-1}$$ (mod $${\boldsymbol{n}}$$ ), we find that $$(\pi[a],\pi[b])$$ is an $$S$$ -symmetry. When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $$\left(a_{i j}\right)$$ . We will meet these in 3.4. We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31]. For any affine algebra $$X_{r}^{(1)}$$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional. There is another general construction of fusion-symmetries, generalising $$C$$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $$\sigma_{\ell}$$ is such that $$0^{(\ell)}$$ is a simple-current $$j$$ — equivalently, that $$\sigma_{\ell}(S_{00}^{2})=S_{00}^{2}$$ — then the permutation is a fusion-symmetry. The simplest example is $$\pi\{-1\}=C$$ . We call $$\pi\{\ell\}$$ a Galois fusion- symmetry. A special case of these was given in [13]. To see that $$\pi\{\ell\}$$ works, note from that $$\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(0)=e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}$$ . Hence and so $$(\pi\{\ell\},\pi\{\ell\}^{-1})$$ is an $$S$$ -symmetry. Incidentally, $$J$$ will always be order 1 or 2 because $$2\,Q_{j}(\lambda)\in\mathbb{Z}$$ for all $$\lambda\in P_{+}$$ . Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings.	<p>$$\mathrm{ch}_{\overline{{{\mu}}}}$$ for $$X_{r}$$ can be written as a polynomial in the fundamental characters $$\mathrm{ch}_{\Lambda_{1}},\ldots,\mathrm{ch}_{\Lambda_{r}}$$ , we know from (2.1b) that $$\Gamma=\{\Lambda_{1},\dots,\Lambda_{r}\}$$ is a fusion-generator for $$X_{r}^{(1)}$$ at any level $$k$$ sufficiently large that $$P_{+}$$ contains all $$\Lambda_{i}$$ (in other words, for any $$k\geq\operatorname*{max}_{i}a_{i}^{\vee}$$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $$X_{r,k}$$ is $$\{\Lambda_{1},\ldots,\Lambda_{r}\}\cap{\cal P}_{+}$$ . Smaller fusion-generators usually exist — for example $$\{\Lambda_{1}\}$$ is a fusion-generator for $$A_{8,k}$$ whenever $$k$$ is even and coprime to 3.</p> <p>2.3. Standard constructions of fusion-symmetries</p> <p>Simple-currents are a large source of fusion-symmetries. Let $$j$$ be any simple-current of order $${\boldsymbol{n}}$$ . Choose any number $$a\in\{0,1,\ldots,n-1\}$$ such that</p> <p>Any solution to this defines a fusion-symmetry $$\lambda\mapsto J^{n a Q_{j}\,(\lambda)}\lambda$$ , which we shall denote $$\pi[a]$$ or $$\pi_{j}[a]$$ . Note that from (2.2b), (2.5b) and (2.5c) that any $$\pi\,=\,\pi[a]$$ , $$a\in\mathbb{Z}$$ , obeys the relation $$N_{\pi\lambda,\pi\mu}^{\pi\nu}=\ N_{\lambda\mu}^{\nu}$$ when $$N_{\lambda\mu}^{\nu}\neq0$$ (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces $$\pi[a]$$ to be a permutation. Choosing $$b\equiv-a\,(n a Q_{j}(j)\!+\!1)^{-1}$$ (mod $${\boldsymbol{n}}$$ ), we find that $$(\pi[a],\pi[b])$$ is an $$S$$ -symmetry.</p> <p>When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $$\left(a_{i j}\right)$$ . We will meet these in 3.4.</p> <p>We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31].</p> <p>For any affine algebra $$X_{r}^{(1)}$$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional.</p> <p>There is another general construction of fusion-symmetries, generalising $$C$$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $$\sigma_{\ell}$$ is such that $$0^{(\ell)}$$ is a simple-current $$j$$ — equivalently, that $$\sigma_{\ell}(S_{00}^{2})=S_{00}^{2}$$ — then the permutation</p> <p>is a fusion-symmetry. The simplest example is $$\pi\{-1\}=C$$ . We call $$\pi\{\ell\}$$ a Galois fusion- symmetry. A special case of these was given in [13]. To see that $$\pi\{\ell\}$$ works, note from</p> <p>that $$\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(0)=e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}$$ . Hence</p> <p>and so $$(\pi\{\ell\},\pi\{\ell\}^{-1})$$ is an $$S$$ -symmetry. Incidentally, $$J$$ will always be order 1 or 2 because $$2\,Q_{j}(\lambda)\in\mathbb{Z}$$ for all $$\lambda\in P_{+}$$ .</p> <p>Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings.</p>	[{"type": "text", "coordinates": [70, 70, 541, 160], "content": "$$\\mathrm{ch}_{\\overline{{{\\mu}}}}$$ for $$X_{r}$$ can be written as a polynomial in the fundamental characters $$\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}$$ ,\nwe know from (2.1b) that $$\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}$$ is a fusion-generator for $$X_{r}^{(1)}$$ at any level $$k$$\nsufficiently large that $$P_{+}$$ contains all $$\\Lambda_{i}$$ (in other words, for any $$k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}$$ ). In fact, it\nis easy to show [18] that a fusion-generator valid for any $$X_{r,k}$$ is $$\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}$$ . Smaller\nfusion-generators usually exist \u2014 for example $$\\{\\Lambda_{1}\\}$$ is a fusion-generator for $$A_{8,k}$$ whenever\n$$k$$ is even and coprime to 3.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [72, 173, 331, 188], "content": "2.3. Standard constructions of fusion-symmetries", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 194, 540, 225], "content": "Simple-currents are a large source of fusion-symmetries. Let $$j$$ be any simple-current\nof order $${\\boldsymbol{n}}$$ . Choose any number $$a\\in\\{0,1,\\ldots,n-1\\}$$ such that", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [239, 239, 372, 252], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [70, 261, 541, 336], "content": "Any solution to this defines a fusion-symmetry $$\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda$$ , which we shall denote $$\\pi[a]$$\nor $$\\pi_{j}[a]$$ . Note that from (2.2b), (2.5b) and (2.5c) that any $$\\pi\\,=\\,\\pi[a]$$ , $$a\\in\\mathbb{Z}$$ , obeys the\nrelation $$N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}$$ when $$N_{\\lambda\\mu}^{\\nu}\\neq0$$ (it would in fact be a fusion-endomorphism \u2014 see\n\u00a72.2); the \u2018gcd\u2019 condition forces $$\\pi[a]$$ to be a permutation. Choosing $$b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}$$\n(mod $${\\boldsymbol{n}}$$ ), we find that $$(\\pi[a],\\pi[b])$$ is an $$S$$ -symmetry.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [71, 336, 541, 378], "content": "When the group of simple-currents is not cyclic, this construction can be generalised\nin a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $$\\left(a_{i j}\\right)$$ .\nWe will meet these in 3.4.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 380, 541, 408], "content": "We will call these simple-current automorphisms. The first examples of these were\nfound by Bernard [2], and were generalised further in [31].", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 409, 541, 453], "content": "For any affine algebra $$X_{r}^{(1)}$$ and any sufficiently high level, we will see in the next\nsection that its fusion-symmetries consist entirely of simple-current automorphisms and\nconjugations. For this reason, any other fusion-symmetry is called exceptional.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 453, 541, 497], "content": "There is another general construction of fusion-symmetries, generalising $$C$$ , although\nit yields few new examples for the affine fusion rings. If the Galois automorphism $$\\sigma_{\\ell}$$ is such\nthat $$0^{(\\ell)}$$ is a simple-current $$j$$ \u2014 equivalently, that $$\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}$$ \u2014 then the permutation", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [257, 509, 354, 524], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [69, 534, 540, 564], "content": "is a fusion-symmetry. The simplest example is $$\\pi\\{-1\\}=C$$ . We call $$\\pi\\{\\ell\\}$$ a Galois fusion-\nsymmetry. A special case of these was given in [13]. To see that $$\\pi\\{\\ell\\}$$ works, note from", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [198, 576, 412, 593], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [70, 603, 254, 619], "content": "that $$\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}$$ . Hence", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [116, 631, 493, 648], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [70, 657, 542, 686], "content": "and so $$(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})$$ is an $$S$$ -symmetry. Incidentally, $$J$$ will always be order 1 or 2 because\n$$2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}$$ for all $$\\lambda\\in P_{+}$$ .", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [70, 687, 542, 715], "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries\nhave analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "block_type": "text", "index": 16}]	[{"type": "inline_equation", "coordinates": [71, 75, 90, 88], "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "score": 0.84, "index": 1}, {"type": "text", "coordinates": [90, 71, 111, 91], "content": " for ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [112, 75, 127, 86], "content": "X_{r}", "score": 0.92, "index": 3}, {"type": "text", "coordinates": [128, 71, 462, 91], "content": " can be written as a polynomial in the fundamental characters ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [462, 74, 537, 87], "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "score": 0.87, "index": 5}, {"type": "text", "coordinates": [537, 71, 542, 91], "content": ",", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [69, 88, 212, 106], "content": "we know from (2.1b) that ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [212, 92, 303, 105], "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "score": 0.93, "index": 8}, {"type": "text", "coordinates": [304, 88, 438, 106], "content": " is a fusion-generator for ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [438, 88, 462, 103], "content": "X_{r}^{(1)}", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [463, 88, 532, 106], "content": "at any level ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [532, 93, 540, 102], "content": "k", "score": 0.84, "index": 12}, {"type": "text", "coordinates": [70, 105, 185, 120], "content": "sufficiently large that ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [186, 108, 201, 119], "content": "P_{+}", "score": 0.92, "index": 14}, {"type": "text", "coordinates": [202, 105, 268, 120], "content": " contains all ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [268, 107, 281, 118], "content": "\\Lambda_{i}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [281, 105, 412, 120], "content": " (in other words, for any ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [412, 105, 478, 119], "content": "k\\geq\\operatorname{max}_{i}a_{i}^{\\vee}", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [479, 105, 541, 120], "content": "). In fact, it", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [70, 119, 362, 135], "content": "is easy to show [18] that a fusion-generator valid for any ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [362, 122, 385, 134], "content": "X_{r,k}", "score": 0.93, "index": 21}, {"type": "text", "coordinates": [385, 119, 399, 135], "content": " is ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [400, 120, 492, 133], "content": "\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}", "score": 0.92, "index": 23}, {"type": "text", "coordinates": [492, 119, 541, 135], "content": ". Smaller", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [70, 133, 310, 149], "content": "fusion-generators usually exist \u2014 for example ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [310, 135, 336, 147], "content": "\\{\\Lambda_{1}\\}", "score": 0.94, "index": 26}, {"type": "text", "coordinates": [336, 133, 465, 149], "content": " is a fusion-generator for ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [465, 135, 487, 148], "content": "A_{8,k}", "score": 0.91, "index": 28}, {"type": "text", "coordinates": [488, 133, 541, 149], "content": " whenever", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [71, 150, 78, 159], "content": "k", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [78, 149, 215, 163], "content": " is even and coprime to 3.", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [72, 177, 330, 189], "content": "2.3. Standard constructions of fusion-symmetries", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [95, 198, 415, 212], "content": "Simple-currents are a large source of fusion-symmetries. Let ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [416, 199, 421, 210], "content": "j", "score": 0.88, "index": 34}, {"type": "text", "coordinates": [421, 198, 540, 212], "content": " be any simple-current", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 211, 116, 226], "content": "of order ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [116, 217, 123, 222], "content": "{\\boldsymbol{n}}", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [124, 211, 239, 226], "content": ". Choose any number ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [240, 212, 344, 225], "content": "a\\in\\{0,1,\\ldots,n-1\\}", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [345, 211, 399, 226], "content": " such that", "score": 1.0, "index": 40}, {"type": "interline_equation", "coordinates": [239, 239, 372, 252], "content": "\\operatorname{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "score": 0.88, "index": 41}, {"type": "text", "coordinates": [70, 262, 317, 280], "content": "Any solution to this defines a fusion-symmetry ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [318, 264, 396, 276], "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "score": 0.93, "index": 43}, {"type": "text", "coordinates": [396, 262, 519, 280], "content": ", which we shall denote ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [519, 266, 540, 279], "content": "\\pi[a]", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [70, 279, 86, 294], "content": "or ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [87, 280, 112, 293], "content": "\\pi_{j}[a]", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [112, 279, 393, 294], "content": ". Note that from (2.2b), (2.5b) and (2.5c) that any ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [393, 280, 441, 293], "content": "\\pi\\,=\\,\\pi[a]", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [441, 279, 448, 294], "content": ", ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [448, 281, 480, 290], "content": "a\\in\\mathbb{Z}", "score": 0.87, "index": 51}, {"type": "text", "coordinates": [481, 279, 541, 294], "content": ", obeys the", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [69, 292, 115, 311], "content": "relation ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [115, 295, 193, 309], "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "score": 0.94, "index": 54}, {"type": "text", "coordinates": [194, 292, 228, 311], "content": " when ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [228, 295, 273, 309], "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "score": 0.94, "index": 56}, {"type": "text", "coordinates": [273, 292, 543, 311], "content": " (it would in fact be a fusion-endomorphism \u2014 see", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [69, 308, 235, 325], "content": "\u00a72.2); the \u2018gcd\u2019 condition forces ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [235, 311, 255, 323], "content": "\\pi[a]", "score": 0.92, "index": 59}, {"type": "text", "coordinates": [255, 308, 418, 325], "content": " to be a permutation. Choosing ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [419, 309, 539, 324], "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [71, 322, 102, 339], "content": "(mod ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [103, 329, 110, 334], "content": "{\\boldsymbol{n}}", "score": 0.82, "index": 63}, {"type": "text", "coordinates": [111, 322, 189, 339], "content": "), we find that ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [189, 324, 244, 337], "content": "(\\pi[a],\\pi[b])", "score": 0.94, "index": 65}, {"type": "text", "coordinates": [244, 322, 275, 339], "content": " is an ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [276, 325, 284, 334], "content": "S", "score": 0.88, "index": 67}, {"type": "text", "coordinates": [284, 322, 343, 339], "content": "-symmetry.", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [95, 338, 541, 353], "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [69, 352, 512, 368], "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [512, 353, 536, 366], "content": "\\left(a_{i j}\\right)", "score": 0.9, "index": 71}, {"type": "text", "coordinates": [537, 352, 540, 368], "content": ".", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [71, 367, 213, 380], "content": "We will meet these in 3.4.", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [94, 380, 541, 396], "content": "We will call these simple-current automorphisms. The first examples of these were", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [72, 396, 376, 410], "content": "found by Bernard [2], and were generalised further in [31].", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [92, 407, 217, 429], "content": "For any affine algebra ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [218, 410, 242, 424], "content": "X_{r}^{(1)}", "score": 0.93, "index": 77}, {"type": "text", "coordinates": [242, 407, 544, 429], "content": "and any sufficiently high level, we will see in the next", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [70, 426, 542, 442], "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [72, 442, 483, 455], "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [94, 454, 476, 470], "content": "There is another general construction of fusion-symmetries, generalising ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [477, 457, 486, 466], "content": "C", "score": 0.88, "index": 82}, {"type": "text", "coordinates": [487, 454, 541, 470], "content": ", although", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [69, 469, 490, 484], "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [491, 474, 502, 482], "content": "\\sigma_{\\ell}", "score": 0.89, "index": 85}, {"type": "text", "coordinates": [502, 469, 541, 484], "content": "is such", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [69, 481, 96, 501], "content": "that ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [97, 483, 115, 495], "content": "0^{(\\ell)}", "score": 0.91, "index": 88}, {"type": "text", "coordinates": [115, 481, 218, 501], "content": " is a simple-current ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [218, 486, 224, 497], "content": "j", "score": 0.85, "index": 90}, {"type": "text", "coordinates": [224, 481, 336, 501], "content": " \u2014 equivalently, that ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [337, 484, 409, 497], "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "score": 0.94, "index": 92}, {"type": "text", "coordinates": [409, 481, 542, 501], "content": " \u2014 then the permutation", "score": 1.0, "index": 93}, {"type": "interline_equation", "coordinates": [257, 509, 354, 524], "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "score": 0.93, "index": 94}, {"type": "text", "coordinates": [68, 536, 317, 553], "content": "is a fusion-symmetry. The simplest example is ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [317, 538, 377, 551], "content": "\\pi\\{-1\\}=C", "score": 0.96, "index": 96}, {"type": "text", "coordinates": [378, 536, 428, 553], "content": ". We call ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [428, 538, 453, 551], "content": "\\pi\\{\\ell\\}", "score": 0.93, "index": 98}, {"type": "text", "coordinates": [453, 536, 541, 553], "content": " a Galois fusion-", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [71, 552, 410, 566], "content": "symmetry. A special case of these was given in [13]. To see that ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [410, 552, 434, 565], "content": "\\pi\\{\\ell\\}", "score": 0.93, "index": 101}, {"type": "text", "coordinates": [435, 552, 528, 566], "content": " works, note from", "score": 1.0, "index": 102}, {"type": "interline_equation", "coordinates": [198, 576, 412, 593], "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "score": 0.89, "index": 103}, {"type": "text", "coordinates": [69, 603, 97, 620], "content": "that ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [97, 606, 212, 620], "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "score": 0.93, "index": 105}, {"type": "text", "coordinates": [212, 603, 254, 620], "content": ". Hence", "score": 1.0, "index": 106}, {"type": "interline_equation", "coordinates": [116, 631, 493, 648], "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "score": 0.91, "index": 107}, {"type": "text", "coordinates": [70, 658, 107, 675], "content": "and so ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [107, 660, 183, 673], "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "score": 0.93, "index": 109}, {"type": "text", "coordinates": [184, 658, 213, 675], "content": " is an ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [213, 661, 222, 670], "content": "S", "score": 0.9, "index": 111}, {"type": "text", "coordinates": [222, 658, 351, 675], "content": "-symmetry. Incidentally, ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [352, 661, 360, 670], "content": "J", "score": 0.88, "index": 113}, {"type": "text", "coordinates": [360, 658, 542, 675], "content": " will always be order 1 or 2 because", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [70, 675, 133, 688], "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [133, 673, 171, 689], "content": " for all ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [172, 676, 209, 687], "content": "\\lambda\\in P_{+}", "score": 0.92, "index": 117}, {"type": "text", "coordinates": [209, 673, 214, 689], "content": ".", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [93, 687, 542, 704], "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "score": 1.0, "index": 119}, {"type": "text", "coordinates": [70, 702, 424, 719], "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "score": 1.0, "index": 120}]	[]	[{"type": "block", "coordinates": [239, 239, 372, 252], "content": "", "caption": ""}, {"type": "block", "coordinates": [257, 509, 354, 524], "content": "", "caption": ""}, {"type": "block", "coordinates": [198, 576, 412, 593], "content": "", "caption": ""}, {"type": "block", "coordinates": [116, 631, 493, 648], "content": "", "caption": ""}, {"type": "inline", "coordinates": [71, 75, 90, 88], "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "caption": ""}, {"type": "inline", "coordinates": [112, 75, 127, 86], "content": "X_{r}", "caption": ""}, {"type": "inline", "coordinates": [462, 74, 537, 87], "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "caption": ""}, {"type": "inline", "coordinates": [212, 92, 303, 105], "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "caption": ""}, {"type": "inline", "coordinates": [438, 88, 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"inline", "coordinates": [416, 199, 421, 210], "content": "j", "caption": ""}, {"type": "inline", "coordinates": [116, 217, 123, 222], "content": "{\\boldsymbol{n}}", "caption": ""}, {"type": "inline", "coordinates": [240, 212, 344, 225], "content": "a\\in\\{0,1,\\ldots,n-1\\}", "caption": ""}, {"type": "inline", "coordinates": [318, 264, 396, 276], "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "caption": ""}, {"type": "inline", "coordinates": [519, 266, 540, 279], "content": "\\pi[a]", "caption": ""}, {"type": "inline", "coordinates": [87, 280, 112, 293], "content": "\\pi_{j}[a]", "caption": ""}, {"type": "inline", "coordinates": [393, 280, 441, 293], "content": "\\pi\\,=\\,\\pi[a]", "caption": ""}, {"type": "inline", "coordinates": [448, 281, 480, 290], "content": "a\\in\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [115, 295, 193, 309], "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "caption": ""}, {"type": 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670], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [352, 661, 360, 670], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [70, 675, 133, 688], "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [172, 676, 209, 687], "content": "\\lambda\\in P_{+}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 7}, {"type": "text", "text": "2.3. Standard constructions of fusion-symmetries ", "page_idx": 7}, {"type": "text", "text": "Simple-currents are a large source of fusion-symmetries. Let $j$ be any simple-current of order ${\\boldsymbol{n}}$ . Choose any number $a\\in\\{0,1,\\ldots,n-1\\}$ such that ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Any solution to this defines a fusion-symmetry $\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda$ , which we shall denote $\\pi[a]$ or $\\pi_{j}[a]$ . Note that from (2.2b), (2.5b) and (2.5c) that any $\\pi\\,=\\,\\pi[a]$ , $a\\in\\mathbb{Z}$ , obeys the relation $N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}$ when $N_{\\lambda\\mu}^{\\nu}\\neq0$ (it would in fact be a fusion-endomorphism \u2014 see \u00a72.2); the \u2018gcd\u2019 condition forces $\\pi[a]$ to be a permutation. Choosing $b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}$ (mod ${\\boldsymbol{n}}$ ), we find that $(\\pi[a],\\pi[b])$ is an $S$ -symmetry. ", "page_idx": 7}, {"type": "text", "text": "When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $\\left(a_{i j}\\right)$ . We will meet these in 3.4. ", "page_idx": 7}, {"type": "text", "text": "We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31]. ", "page_idx": 7}, {"type": "text", "text": "For any affine algebra $X_{r}^{(1)}$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional. ", "page_idx": 7}, {"type": "text", "text": "There is another general construction of fusion-symmetries, generalising $C$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $\\sigma_{\\ell}$ is such that $0^{(\\ell)}$ is a simple-current $j$ \u2014 equivalently, that $\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}$ \u2014 then the permutation ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "is a fusion-symmetry. The simplest example is $\\pi\\{-1\\}=C$ . We call $\\pi\\{\\ell\\}$ a Galois fusionsymmetry. A special case of these was given in [13]. To see that $\\pi\\{\\ell\\}$ works, note from ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "that $\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}$ . Hence ", "page_idx": 7}, {"type": "equation", "text": "$$\nS_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and so $(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})$ is an $S$ -symmetry. Incidentally, $J$ will always be order 1 or 2 because $2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}$ for all $\\lambda\\in P_{+}$ . ", "page_idx": 7}, {"type": "text", "text": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings. 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Standard constructions of fusion-symmetries", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 194, 540, 225], "lines": [{"bbox": [95, 198, 540, 212], "spans": [{"bbox": [95, 198, 415, 212], "score": 1.0, "content": "Simple-currents are a large source of fusion-symmetries. Let ", "type": "text"}, {"bbox": [416, 199, 421, 210], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [421, 198, 540, 212], "score": 1.0, "content": " be any simple-current", "type": "text"}], "index": 7}, {"bbox": [70, 211, 399, 226], "spans": [{"bbox": [70, 211, 116, 226], "score": 1.0, "content": "of order ", "type": "text"}, {"bbox": [116, 217, 123, 222], "score": 0.89, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [124, 211, 239, 226], "score": 1.0, "content": ". Choose any number ", "type": "text"}, {"bbox": [240, 212, 344, 225], "score": 0.93, "content": "a\\in\\{0,1,\\ldots,n-1\\}", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [345, 211, 399, 226], "score": 1.0, "content": " such that", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 261, 541, 336], "lines": [{"bbox": [70, 262, 540, 280], "spans": [{"bbox": [70, 262, 317, 280], "score": 1.0, "content": "Any solution to this defines a fusion-symmetry ", "type": "text"}, {"bbox": [318, 264, 396, 276], "score": 0.93, "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [396, 262, 519, 280], "score": 1.0, "content": ", which we shall denote ", "type": "text"}, {"bbox": [519, 266, 540, 279], "score": 0.91, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}], "index": 10}, {"bbox": [70, 279, 541, 294], "spans": [{"bbox": [70, 279, 86, 294], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [87, 280, 112, 293], "score": 0.93, "content": "\\pi_{j}[a]", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [112, 279, 393, 294], "score": 1.0, "content": ". Note that from (2.2b), (2.5b) and (2.5c) that any ", "type": "text"}, {"bbox": [393, 280, 441, 293], "score": 0.92, "content": "\\pi\\,=\\,\\pi[a]", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [441, 279, 448, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [448, 281, 480, 290], "score": 0.87, "content": "a\\in\\mathbb{Z}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [481, 279, 541, 294], "score": 1.0, "content": ", obeys the", "type": "text"}], "index": 11}, {"bbox": [69, 292, 543, 311], "spans": [{"bbox": [69, 292, 115, 311], "score": 1.0, "content": "relation ", "type": "text"}, {"bbox": [115, 295, 193, 309], "score": 0.94, "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [194, 292, 228, 311], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [228, 295, 273, 309], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [273, 292, 543, 311], "score": 1.0, "content": " (it would in fact be a fusion-endomorphism \u2014 see", "type": "text"}], "index": 12}, {"bbox": [69, 308, 539, 325], "spans": [{"bbox": [69, 308, 235, 325], "score": 1.0, "content": "\u00a72.2); the \u2018gcd\u2019 condition forces ", "type": "text"}, {"bbox": [235, 311, 255, 323], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [255, 308, 418, 325], "score": 1.0, "content": " to be a permutation. Choosing ", "type": "text"}, {"bbox": [419, 309, 539, 324], "score": 0.92, "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "type": "inline_equation", "height": 15, "width": 120}], "index": 13}, {"bbox": [71, 322, 343, 339], "spans": [{"bbox": [71, 322, 102, 339], "score": 1.0, "content": "(mod ", "type": "text"}, {"bbox": [103, 329, 110, 334], "score": 0.82, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [111, 322, 189, 339], "score": 1.0, "content": "), we find that ", "type": "text"}, {"bbox": [189, 324, 244, 337], "score": 0.94, "content": "(\\pi[a],\\pi[b])", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [244, 322, 275, 339], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [276, 325, 284, 334], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [284, 322, 343, 339], "score": 1.0, "content": "-symmetry.", "type": "text"}], "index": 14}], "index": 12}, {"type": "text", "bbox": [71, 336, 541, 378], "lines": [{"bbox": [95, 338, 541, 353], "spans": [{"bbox": [95, 338, 541, 353], "score": 1.0, "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "type": "text"}], "index": 15}, {"bbox": [69, 352, 540, 368], "spans": [{"bbox": [69, 352, 512, 368], "score": 1.0, "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix ", "type": "text"}, {"bbox": [512, 353, 536, 366], "score": 0.9, "content": "\\left(a_{i j}\\right)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [537, 352, 540, 368], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [71, 367, 213, 380], "spans": [{"bbox": [71, 367, 213, 380], "score": 1.0, "content": "We will meet these in 3.4.", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [70, 380, 541, 408], "lines": [{"bbox": [94, 380, 541, 396], "spans": [{"bbox": [94, 380, 541, 396], "score": 1.0, "content": "We will call these simple-current automorphisms. The first examples of these were", "type": "text"}], "index": 18}, {"bbox": [72, 396, 376, 410], "spans": [{"bbox": [72, 396, 376, 410], "score": 1.0, "content": "found by Bernard [2], and were generalised further in [31].", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "text", "bbox": [70, 409, 541, 453], "lines": [{"bbox": [92, 407, 544, 429], "spans": [{"bbox": [92, 407, 217, 429], "score": 1.0, "content": "For any affine algebra ", "type": "text"}, {"bbox": [218, 410, 242, 424], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [242, 407, 544, 429], "score": 1.0, "content": "and any sufficiently high level, we will see in the next", "type": "text"}], "index": 20}, {"bbox": [70, 426, 542, 442], "spans": [{"bbox": [70, 426, 542, 442], "score": 1.0, "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "type": "text"}], "index": 21}, {"bbox": [72, 442, 483, 455], "spans": [{"bbox": [72, 442, 483, 455], "score": 1.0, "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [70, 453, 541, 497], "lines": [{"bbox": [94, 454, 541, 470], "spans": [{"bbox": [94, 454, 476, 470], "score": 1.0, "content": "There is another general construction of fusion-symmetries, generalising ", "type": "text"}, {"bbox": [477, 457, 486, 466], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [487, 454, 541, 470], "score": 1.0, "content": ", although", "type": "text"}], "index": 23}, {"bbox": [69, 469, 541, 484], "spans": [{"bbox": [69, 469, 490, 484], "score": 1.0, "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism ", "type": "text"}, {"bbox": [491, 474, 502, 482], "score": 0.89, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [502, 469, 541, 484], "score": 1.0, "content": "is such", "type": "text"}], "index": 24}, {"bbox": [69, 481, 542, 501], "spans": [{"bbox": [69, 481, 96, 501], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 483, 115, 495], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [115, 481, 218, 501], "score": 1.0, "content": " is a simple-current ", "type": "text"}, {"bbox": [218, 486, 224, 497], "score": 0.85, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [224, 481, 336, 501], "score": 1.0, "content": " \u2014 equivalently, that ", "type": "text"}, {"bbox": [337, 484, 409, 497], "score": 0.94, "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [409, 481, 542, 501], "score": 1.0, "content": " \u2014 then the permutation", "type": "text"}], "index": 25}], "index": 24}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [69, 534, 540, 564], "lines": [{"bbox": [68, 536, 541, 553], "spans": [{"bbox": [68, 536, 317, 553], "score": 1.0, "content": "is a fusion-symmetry. The simplest example is ", "type": "text"}, {"bbox": [317, 538, 377, 551], "score": 0.96, "content": "\\pi\\{-1\\}=C", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [378, 536, 428, 553], "score": 1.0, "content": ". We call ", "type": "text"}, {"bbox": [428, 538, 453, 551], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [453, 536, 541, 553], "score": 1.0, "content": " a Galois fusion-", "type": "text"}], "index": 27}, {"bbox": [71, 552, 528, 566], "spans": [{"bbox": [71, 552, 410, 566], "score": 1.0, "content": "symmetry. A special case of these was given in [13]. To see that ", "type": "text"}, {"bbox": [410, 552, 434, 565], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [435, 552, 528, 566], "score": 1.0, "content": " works, note from", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [70, 603, 254, 619], "lines": [{"bbox": [69, 603, 254, 620], "spans": [{"bbox": [69, 603, 97, 620], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 606, 212, 620], "score": 0.93, "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "type": "inline_equation", "height": 14, "width": 115}, {"bbox": [212, 603, 254, 620], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [70, 657, 542, 686], "lines": [{"bbox": [70, 658, 542, 675], "spans": [{"bbox": [70, 658, 107, 675], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [107, 660, 183, 673], "score": 0.93, "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [184, 658, 213, 675], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [213, 661, 222, 670], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [222, 658, 351, 675], "score": 1.0, "content": "-symmetry. Incidentally, ", "type": "text"}, {"bbox": [352, 661, 360, 670], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 658, 542, 675], "score": 1.0, "content": " will always be order 1 or 2 because", "type": "text"}], "index": 32}, {"bbox": [70, 673, 214, 689], "spans": [{"bbox": [70, 675, 133, 688], "score": 0.92, "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [133, 673, 171, 689], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [172, 676, 209, 687], "score": 0.92, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [209, 673, 214, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "text", "bbox": [70, 687, 542, 715], "lines": [{"bbox": [93, 687, 542, 704], "spans": [{"bbox": [93, 687, 542, 704], "score": 1.0, "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "type": "text"}], "index": 34}, {"bbox": [70, 702, 424, 719], "spans": [{"bbox": [70, 702, 424, 719], "score": 1.0, "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "type": "text"}], "index": 35}], "index": 34.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 160], "lines": [], "index": 2.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 71, 542, 163], "lines_deleted": true}, {"type": "text", "bbox": [72, 173, 331, 188], "lines": [{"bbox": [72, 177, 330, 189], "spans": [{"bbox": [72, 177, 330, 189], "score": 1.0, "content": "2.3. Standard constructions of fusion-symmetries", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [72, 177, 330, 189]}, {"type": "text", "bbox": [70, 194, 540, 225], "lines": [{"bbox": [95, 198, 540, 212], "spans": [{"bbox": [95, 198, 415, 212], "score": 1.0, "content": "Simple-currents are a large source of fusion-symmetries. Let ", "type": "text"}, {"bbox": [416, 199, 421, 210], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [421, 198, 540, 212], "score": 1.0, "content": " be any simple-current", "type": "text"}], "index": 7}, {"bbox": [70, 211, 399, 226], "spans": [{"bbox": [70, 211, 116, 226], "score": 1.0, "content": "of order ", "type": "text"}, {"bbox": [116, 217, 123, 222], "score": 0.89, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [124, 211, 239, 226], "score": 1.0, "content": ". Choose any number ", "type": "text"}, {"bbox": [240, 212, 344, 225], "score": 0.93, "content": "a\\in\\{0,1,\\ldots,n-1\\}", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [345, 211, 399, 226], "score": 1.0, "content": " such that", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 198, 540, 226]}, {"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 261, 541, 336], "lines": [{"bbox": [70, 262, 540, 280], "spans": [{"bbox": [70, 262, 317, 280], "score": 1.0, "content": "Any solution to this defines a fusion-symmetry ", "type": "text"}, {"bbox": [318, 264, 396, 276], "score": 0.93, "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [396, 262, 519, 280], "score": 1.0, "content": ", which we shall denote ", "type": "text"}, {"bbox": [519, 266, 540, 279], "score": 0.91, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}], "index": 10}, {"bbox": [70, 279, 541, 294], "spans": [{"bbox": [70, 279, 86, 294], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [87, 280, 112, 293], "score": 0.93, "content": "\\pi_{j}[a]", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [112, 279, 393, 294], "score": 1.0, "content": ". Note that from (2.2b), (2.5b) and (2.5c) that any ", "type": "text"}, {"bbox": [393, 280, 441, 293], "score": 0.92, "content": "\\pi\\,=\\,\\pi[a]", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [441, 279, 448, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [448, 281, 480, 290], "score": 0.87, "content": "a\\in\\mathbb{Z}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [481, 279, 541, 294], "score": 1.0, "content": ", obeys the", "type": "text"}], "index": 11}, {"bbox": [69, 292, 543, 311], "spans": [{"bbox": [69, 292, 115, 311], "score": 1.0, "content": "relation ", "type": "text"}, {"bbox": [115, 295, 193, 309], "score": 0.94, "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [194, 292, 228, 311], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [228, 295, 273, 309], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [273, 292, 543, 311], "score": 1.0, "content": " (it would in fact be a fusion-endomorphism \u2014 see", "type": "text"}], "index": 12}, {"bbox": [69, 308, 539, 325], "spans": [{"bbox": [69, 308, 235, 325], "score": 1.0, "content": "\u00a72.2); the \u2018gcd\u2019 condition forces ", "type": "text"}, {"bbox": [235, 311, 255, 323], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [255, 308, 418, 325], "score": 1.0, "content": " to be a permutation. Choosing ", "type": "text"}, {"bbox": [419, 309, 539, 324], "score": 0.92, "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "type": "inline_equation", "height": 15, "width": 120}], "index": 13}, {"bbox": [71, 322, 343, 339], "spans": [{"bbox": [71, 322, 102, 339], "score": 1.0, "content": "(mod ", "type": "text"}, {"bbox": [103, 329, 110, 334], "score": 0.82, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [111, 322, 189, 339], "score": 1.0, "content": "), we find that ", "type": "text"}, {"bbox": [189, 324, 244, 337], "score": 0.94, "content": "(\\pi[a],\\pi[b])", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [244, 322, 275, 339], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [276, 325, 284, 334], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [284, 322, 343, 339], "score": 1.0, "content": "-symmetry.", "type": "text"}], "index": 14}], "index": 12, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 262, 543, 339]}, {"type": "text", "bbox": [71, 336, 541, 378], "lines": [{"bbox": [95, 338, 541, 353], "spans": [{"bbox": [95, 338, 541, 353], "score": 1.0, "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "type": "text"}], "index": 15}, {"bbox": [69, 352, 540, 368], "spans": [{"bbox": [69, 352, 512, 368], "score": 1.0, "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix ", "type": "text"}, {"bbox": [512, 353, 536, 366], "score": 0.9, "content": "\\left(a_{i j}\\right)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [537, 352, 540, 368], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [71, 367, 213, 380], "spans": [{"bbox": [71, 367, 213, 380], "score": 1.0, "content": "We will meet these in 3.4.", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 338, 541, 380]}, {"type": "text", "bbox": [70, 380, 541, 408], "lines": [{"bbox": [94, 380, 541, 396], "spans": [{"bbox": [94, 380, 541, 396], "score": 1.0, "content": "We will call these simple-current automorphisms. The first examples of these were", "type": "text"}], "index": 18}, {"bbox": [72, 396, 376, 410], "spans": [{"bbox": [72, 396, 376, 410], "score": 1.0, "content": "found by Bernard [2], and were generalised further in [31].", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [72, 380, 541, 410]}, {"type": "text", "bbox": [70, 409, 541, 453], "lines": [{"bbox": [92, 407, 544, 429], "spans": [{"bbox": [92, 407, 217, 429], "score": 1.0, "content": "For any affine algebra ", "type": "text"}, {"bbox": [218, 410, 242, 424], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [242, 407, 544, 429], "score": 1.0, "content": "and any sufficiently high level, we will see in the next", "type": "text"}], "index": 20}, {"bbox": [70, 426, 542, 442], "spans": [{"bbox": [70, 426, 542, 442], "score": 1.0, "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "type": "text"}], "index": 21}, {"bbox": [72, 442, 483, 455], "spans": [{"bbox": [72, 442, 483, 455], "score": 1.0, "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 407, 544, 455]}, {"type": "text", "bbox": [70, 453, 541, 497], "lines": [{"bbox": [94, 454, 541, 470], "spans": [{"bbox": [94, 454, 476, 470], "score": 1.0, "content": "There is another general construction of fusion-symmetries, generalising ", "type": "text"}, {"bbox": [477, 457, 486, 466], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [487, 454, 541, 470], "score": 1.0, "content": ", although", "type": "text"}], "index": 23}, {"bbox": [69, 469, 541, 484], "spans": [{"bbox": [69, 469, 490, 484], "score": 1.0, "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism ", "type": "text"}, {"bbox": [491, 474, 502, 482], "score": 0.89, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [502, 469, 541, 484], "score": 1.0, "content": "is such", "type": "text"}], "index": 24}, {"bbox": [69, 481, 542, 501], "spans": [{"bbox": [69, 481, 96, 501], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 483, 115, 495], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [115, 481, 218, 501], "score": 1.0, "content": " is a simple-current ", "type": "text"}, {"bbox": [218, 486, 224, 497], "score": 0.85, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [224, 481, 336, 501], "score": 1.0, "content": " \u2014 equivalently, that ", "type": "text"}, {"bbox": [337, 484, 409, 497], "score": 0.94, "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [409, 481, 542, 501], "score": 1.0, "content": " \u2014 then the permutation", "type": "text"}], "index": 25}], "index": 24, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 454, 542, 501]}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 534, 540, 564], "lines": [{"bbox": [68, 536, 541, 553], "spans": [{"bbox": [68, 536, 317, 553], "score": 1.0, "content": "is a fusion-symmetry. The simplest example is ", "type": "text"}, {"bbox": [317, 538, 377, 551], "score": 0.96, "content": "\\pi\\{-1\\}=C", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [378, 536, 428, 553], "score": 1.0, "content": ". We call ", "type": "text"}, {"bbox": [428, 538, 453, 551], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [453, 536, 541, 553], "score": 1.0, "content": " a Galois fusion-", "type": "text"}], "index": 27}, {"bbox": [71, 552, 528, 566], "spans": [{"bbox": [71, 552, 410, 566], "score": 1.0, "content": "symmetry. A special case of these was given in [13]. To see that ", "type": "text"}, {"bbox": [410, 552, 434, 565], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [435, 552, 528, 566], "score": 1.0, "content": " works, note from", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [68, 536, 541, 566]}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 603, 254, 619], "lines": [{"bbox": [69, 603, 254, 620], "spans": [{"bbox": [69, 603, 97, 620], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 606, 212, 620], "score": 0.93, "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "type": "inline_equation", "height": 14, "width": 115}, {"bbox": [212, 603, 254, 620], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 30}], "index": 30, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 603, 254, 620]}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 657, 542, 686], "lines": [{"bbox": [70, 658, 542, 675], "spans": [{"bbox": [70, 658, 107, 675], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [107, 660, 183, 673], "score": 0.93, "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [184, 658, 213, 675], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [213, 661, 222, 670], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [222, 658, 351, 675], "score": 1.0, "content": "-symmetry. Incidentally, ", "type": "text"}, {"bbox": [352, 661, 360, 670], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 658, 542, 675], "score": 1.0, "content": " will always be order 1 or 2 because", "type": "text"}], "index": 32}, {"bbox": [70, 673, 214, 689], "spans": [{"bbox": [70, 675, 133, 688], "score": 0.92, "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [133, 673, 171, 689], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [172, 676, 209, 687], "score": 0.92, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [209, 673, 214, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 658, 542, 689]}, {"type": "text", "bbox": [70, 687, 542, 715], "lines": [{"bbox": [93, 687, 542, 704], "spans": [{"bbox": [93, 687, 542, 704], "score": 1.0, "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "type": "text"}], "index": 34}, {"bbox": [70, 702, 424, 719], "spans": [{"bbox": [70, 702, 424, 719], "score": 1.0, "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "type": "text"}], "index": 35}], "index": 34.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 687, 542, 719]}]}
0002044v1	3	Charge-conjugation is the order 2 permutation of $$P_{+}$$ given by $$C\lambda\,=\,^{t}\lambda$$ , the weight contragredient to $$\lambda$$ . For instance $$C0=0$$ . It has the basic property that and $$S^{2}=C$$ . $$C$$ corresponds to a symmetry of the (unextended) Dynkin diagram of $$X_{r}$$ , as we will see next section. Related to $$C$$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $$X_{r}^{(1)}$$ with nontrivial conjugations other than charge- conjugation are $$D_{e v e n}^{(1)}$$ . Another important symmetry of the matrix $$S$$ is called simple-currents. Any weight $$j\in P_{+}$$ with q-dimension $$\mathcal{D}(j)=1$$ , is called a simple-current. To any such weight $$j$$ is associated a permutation $$J$$ of $$P_{+}$$ and a function $$Q_{j}:P_{+}\to\mathbb{Q}$$ , such that $$J0=j$$ and The simple-currents form an abelian group, given by composition of the permutations $$J$$ . All simple-currents for the affine algebras were classified in [12], and with one unimpor- tant exception ( $${E}_{8}^{(1)}$$ at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of $$X_{r}^{(1)}$$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $$W_{0}^{+}$$ . Evaluating $$S_{J\lambda,j^{\prime}}$$ in two ways gives the useful and hence the reciprocity $$Q_{j}(j^{\prime})=Q_{j^{\prime}}(j)$$ . For each $$X_{r}$$ , the inner products $$(\lambda\|\mu)$$ of weights are rational; let $$N$$ denote the least common denominator. E.g. for $$A_{r}$$ this is $$N=r+1$$ , while for $$E_{8}$$ it is $$N=1$$ . Choose any integer $$\ell$$ coprime to $$\kappa N$$ . Then for any $$\lambda\in P_{+}$$ there is a unique weight $$\lambda^{(\ell)}\in P_{+}$$ , coroot $$\alpha$$ , and (finite) Weyl element $$\omega$$ such that This is simply the statement that the affine Weyl orbit of $$\ell\left(\lambda+\rho\right)$$ intersects the set $$P_{+}+\rho$$ at precisely one point (namely $$\lambda^{(\ell)}+\rho)$$ . Write $$\epsilon_{\ell}^{\prime}(\lambda)=\operatorname*{det}\omega=\pm1$$ . Then [16] This has an obvious interpretation as a Galois automorphism [4]: the field generated over $$\mathbb{Q}$$ by all entries $$S_{\lambda\mu}$$ lies in the cyclotomic field $$\mathbb{Q}[\xi_{4N\kappa}]$$ where $$\xi_{n}$$ denotes the root of unity $$\exp[2\pi\mathrm{i}/n]$$ ; for any $$\sigma_{\ell}\in{\mathrm{Gal}}(\mathbb{Q}[\xi_{4N\kappa}]/\mathbb{Q})\cong\mathbb{Z}_{4N\kappa}^{\times}$$ , there will be a function $$\epsilon_{\ell}:P_{+}\to\{\pm1\}$$ such that	<p>Charge-conjugation is the order 2 permutation of $$P_{+}$$ given by $$C\lambda\,=\,^{t}\lambda$$ , the weight contragredient to $$\lambda$$ . For instance $$C0=0$$ . It has the basic property that</p> <p>and $$S^{2}=C$$ . $$C$$ corresponds to a symmetry of the (unextended) Dynkin diagram of $$X_{r}$$ , as we will see next section.</p> <p>Related to $$C$$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $$X_{r}^{(1)}$$ with nontrivial conjugations other than charge- conjugation are $$D_{e v e n}^{(1)}$$ .</p> <p>Another important symmetry of the matrix $$S$$ is called simple-currents. Any weight $$j\in P_{+}$$ with q-dimension $$\mathcal{D}(j)=1$$ , is called a simple-current. To any such weight $$j$$ is associated a permutation $$J$$ of $$P_{+}$$ and a function $$Q_{j}:P_{+}\to\mathbb{Q}$$ , such that $$J0=j$$ and</p> <p>The simple-currents form an abelian group, given by composition of the permutations $$J$$ .</p> <p>All simple-currents for the affine algebras were classified in [12], and with one unimpor- tant exception ( $${E}_{8}^{(1)}$$ at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of $$X_{r}^{(1)}$$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $$W_{0}^{+}$$ .</p> <p>Evaluating $$S_{J\lambda,j^{\prime}}$$ in two ways gives the useful</p> <p>and hence the reciprocity $$Q_{j}(j^{\prime})=Q_{j^{\prime}}(j)$$ .</p> <p>For each $$X_{r}$$ , the inner products $$(\lambda\|\mu)$$ of weights are rational; let $$N$$ denote the least common denominator. E.g. for $$A_{r}$$ this is $$N=r+1$$ , while for $$E_{8}$$ it is $$N=1$$ . Choose any integer $$\ell$$ coprime to $$\kappa N$$ . Then for any $$\lambda\in P_{+}$$ there is a unique weight $$\lambda^{(\ell)}\in P_{+}$$ , coroot $$\alpha$$ , and (finite) Weyl element $$\omega$$ such that</p> <p>This is simply the statement that the affine Weyl orbit of $$\ell\left(\lambda+\rho\right)$$ intersects the set $$P_{+}+\rho$$ at precisely one point (namely $$\lambda^{(\ell)}+\rho)$$ . Write $$\epsilon_{\ell}^{\prime}(\lambda)=\operatorname*{det}\omega=\pm1$$ . Then [16]</p> <p>This has an obvious interpretation as a Galois automorphism [4]: the field generated over $$\mathbb{Q}$$ by all entries $$S_{\lambda\mu}$$ lies in the cyclotomic field $$\mathbb{Q}[\xi_{4N\kappa}]$$ where $$\xi_{n}$$ denotes the root of unity $$\exp[2\pi\mathrm{i}/n]$$ ; for any $$\sigma_{\ell}\in{\mathrm{Gal}}(\mathbb{Q}[\xi_{4N\kappa}]/\mathbb{Q})\cong\mathbb{Z}_{4N\kappa}^{\times}$$ , there will be a function $$\epsilon_{\ell}:P_{+}\to\{\pm1\}$$ such that</p>	[{"type": "text", "coordinates": [70, 70, 540, 100], "content": "Charge-conjugation is the order 2 permutation of $$P_{+}$$ given by $$C\\lambda\\,=\\,^{t}\\lambda$$ , the weight\ncontragredient to $$\\lambda$$ . For instance $$C0=0$$ . It has the basic property that", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [250, 117, 361, 132], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 143, 540, 172], "content": "and $$S^{2}=C$$ . $$C$$ corresponds to a symmetry of the (unextended) Dynkin diagram of $$X_{r}$$ ,\nas we will see next section.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 173, 541, 219], "content": "Related to $$C$$ are all the other symmetries of the unextended Dynkin diagram. We\ncall these conjugations. The only $$X_{r}^{(1)}$$ with nontrivial conjugations other than charge-\nconjugation are $$D_{e v e n}^{(1)}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 219, 541, 263], "content": "Another important symmetry of the matrix $$S$$ is called simple-currents. Any weight\n$$j\\in P_{+}$$ with q-dimension $$\\mathcal{D}(j)=1$$ , is called a simple-current. To any such weight $$j$$ is\nassociated a permutation $$J$$ of $$P_{+}$$ and a function $$Q_{j}:P_{+}\\to\\mathbb{Q}$$ , such that $$J0=j$$ and", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [235, 278, 376, 293], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [70, 306, 538, 320], "content": "The simple-currents form an abelian group, given by composition of the permutations $$J$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 321, 541, 397], "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-\ntant exception ( $${E}_{8}^{(1)}$$ at level 2) correspond to symmetries of the extended Coxeter\u2013Dynkin\ndiagram of $$X_{r}^{(1)}$$ . The simplest proof would use the methods of Proposition 4.1 below. For\na more intrinsically algebraic interpretation of these simple-currents, see [25] where their\ngroup is denoted $$W_{0}^{+}$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [94, 397, 338, 412], "content": "Evaluating $$S_{J\\lambda,j^{\\prime}}$$ in two ways gives the useful", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [201, 426, 409, 443], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [70, 455, 294, 469], "content": "and hence the reciprocity $$Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 470, 541, 527], "content": "For each $$X_{r}$$ , the inner products $$(\\lambda\|\\mu)$$ of weights are rational; let $$N$$ denote the least\ncommon denominator. E.g. for $$A_{r}$$ this is $$N=r+1$$ , while for $$E_{8}$$ it is $$N=1$$ . Choose any\ninteger $$\\ell$$ coprime to $$\\kappa N$$ . Then for any $$\\lambda\\in P_{+}$$ there is a unique weight $$\\lambda^{(\\ell)}\\in P_{+}$$ , coroot\n$$\\alpha$$ , and (finite) Weyl element $$\\omega$$ such that", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [228, 541, 383, 557], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [70, 570, 541, 600], "content": "This is simply the statement that the affine Weyl orbit of $$\\ell\\left(\\lambda+\\rho\\right)$$ intersects the set $$P_{+}+\\rho$$\nat precisely one point (namely $$\\lambda^{(\\ell)}+\\rho)$$ . Write $$\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1$$ . Then [16]", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [236, 614, 374, 631], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [68, 642, 541, 699], "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over\n$$\\mathbb{Q}$$ by all entries $$S_{\\lambda\\mu}$$ lies in the cyclotomic field $$\\mathbb{Q}[\\xi_{4N\\kappa}]$$ where $$\\xi_{n}$$ denotes the root of unity\n$$\\exp[2\\pi\\mathrm{i}/n]$$ ; for any $$\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}$$ , there will be a function $$\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}$$\nsuch that", "block_type": "text", "index": 16}, {"type": "interline_equation", "coordinates": [204, 701, 407, 718], "content": "", "block_type": "interline_equation", "index": 17}]	[{"type": "text", "coordinates": [95, 72, 360, 90], "content": "Charge-conjugation is the order 2 permutation of ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [360, 75, 376, 87], "content": "P_{+}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [376, 72, 429, 90], "content": " given by ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [430, 74, 476, 84], "content": "C\\lambda\\,=\\,^{t}\\lambda", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [477, 72, 542, 90], "content": ", the weight", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 88, 165, 104], "content": "contragredient to ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [165, 90, 172, 99], "content": "\\lambda", "score": 0.88, "index": 7}, {"type": "text", "coordinates": [173, 88, 248, 104], "content": ". For instance ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [248, 90, 286, 99], "content": "C0=0", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [286, 88, 452, 104], "content": ". It has the basic property that", "score": 1.0, "index": 10}, {"type": "interline_equation", "coordinates": [250, 117, 361, 132], "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{}", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [70, 145, 95, 162], "content": "and ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [95, 146, 136, 156], "content": "S^{2}=C", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [136, 145, 145, 162], "content": ". ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [146, 148, 155, 157], "content": "C", "score": 0.89, "index": 15}, {"type": "text", "coordinates": [156, 145, 520, 162], "content": " corresponds to a symmetry of the (unextended) Dynkin diagram of ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [520, 148, 536, 159], "content": "X_{r}", "score": 0.91, "index": 17}, {"type": "text", "coordinates": [537, 145, 540, 162], "content": ",", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [71, 161, 212, 174], "content": "as we will see next section.", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [94, 174, 155, 190], "content": "Related to ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [155, 177, 165, 185], "content": "C", "score": 0.9, "index": 21}, {"type": "text", "coordinates": [165, 174, 540, 190], "content": " are all the other symmetries of the unextended Dynkin diagram. We", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [68, 186, 256, 209], "content": "call these conjugations. The only ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [256, 188, 280, 203], "content": "X_{r}^{(1)}", "score": 0.93, "index": 24}, {"type": "text", "coordinates": [281, 186, 542, 209], "content": "with nontrivial conjugations other than charge-", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [70, 203, 156, 226], "content": "conjugation are ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [156, 204, 187, 219], "content": "D_{e v e n}^{(1)}", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [187, 203, 192, 226], "content": ".", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [95, 221, 330, 236], "content": "Another important symmetry of the matrix ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [330, 223, 339, 232], "content": "S", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [339, 221, 541, 236], "content": " is called simple-currents. Any weight", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [71, 238, 110, 249], "content": "j\\in P_{+}", "score": 0.93, "index": 32}, {"type": "text", "coordinates": [110, 236, 210, 250], "content": " with q-dimension ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [210, 237, 260, 249], "content": "\\mathcal{D}(j)=1", "score": 0.94, "index": 34}, {"type": "text", "coordinates": [261, 236, 520, 250], "content": ", is called a simple-current. To any such weight ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [520, 238, 527, 249], "content": "j", "score": 0.88, "index": 36}, {"type": "text", "coordinates": [527, 236, 541, 250], "content": " is", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [71, 250, 206, 264], "content": "associated a permutation ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [207, 252, 214, 261], "content": "J", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [215, 250, 231, 264], "content": " of ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [232, 252, 248, 263], "content": "P_{+}", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [248, 250, 331, 264], "content": " and a function ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [332, 252, 400, 264], "content": "Q_{j}:P_{+}\\to\\mathbb{Q}", "score": 0.93, "index": 43}, {"type": "text", "coordinates": [401, 250, 459, 264], "content": ", such that ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [460, 252, 495, 263], "content": "J0=j", "score": 0.94, "index": 45}, {"type": "text", "coordinates": [496, 250, 520, 264], "content": " and", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [235, 278, 376, 293], "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [71, 308, 525, 323], "content": "The simple-currents form an abelian group, given by composition of the permutations ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [525, 310, 533, 319], "content": "J", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [533, 308, 537, 323], "content": ".", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [94, 322, 541, 338], "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [69, 334, 154, 358], "content": "tant exception (", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [154, 336, 176, 353], "content": "{E}_{8}^{(1)}", "score": 0.92, "index": 53}, {"type": "text", "coordinates": [177, 334, 541, 358], "content": "at level 2) correspond to symmetries of the extended Coxeter\u2013Dynkin", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [69, 353, 130, 372], "content": "diagram of ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [131, 354, 154, 368], "content": "X_{r}^{(1)}", "score": 0.93, "index": 56}, {"type": "text", "coordinates": [155, 353, 542, 372], "content": ". The simplest proof would use the methods of Proposition 4.1 below. For", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [70, 371, 540, 385], "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [69, 384, 162, 402], "content": "group is denoted ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [162, 385, 183, 398], "content": "W_{0}^{+}", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [184, 384, 188, 402], "content": ".", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [95, 399, 155, 414], "content": "Evaluating ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [155, 401, 185, 413], "content": "S_{J\\lambda,j^{\\prime}}", "score": 0.93, "index": 63}, {"type": "text", "coordinates": [186, 399, 336, 414], "content": " in two ways gives the useful", "score": 1.0, "index": 64}, {"type": "interline_equation", "coordinates": [201, 426, 409, 443], "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "score": 0.89, "index": 65}, {"type": "text", "coordinates": [70, 457, 208, 472], "content": "and hence the reciprocity ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [208, 458, 289, 471], "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [290, 457, 293, 472], "content": ".", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [94, 471, 143, 487], "content": "For each ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [144, 473, 159, 484], "content": "X_{r}", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [159, 471, 267, 487], "content": ", the inner products ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [268, 473, 295, 485], "content": "(\\lambda\|\\mu)", "score": 0.93, "index": 72}, {"type": "text", "coordinates": [295, 471, 441, 487], "content": " of weights are rational; let ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [441, 473, 452, 482], "content": "N", "score": 0.89, "index": 74}, {"type": "text", "coordinates": [452, 471, 541, 487], "content": " denote the least", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [69, 484, 235, 502], "content": "common denominator. E.g. for ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [235, 488, 250, 498], "content": "A_{r}", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [250, 484, 288, 502], "content": " this is ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [288, 488, 341, 497], "content": "N=r+1", "score": 0.9, "index": 79}, {"type": "text", "coordinates": [341, 484, 396, 502], "content": ", while for ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [396, 487, 411, 498], "content": "E_{8}", "score": 0.86, "index": 81}, {"type": "text", "coordinates": [411, 484, 437, 502], "content": " it is ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [438, 488, 471, 497], "content": "N=1", "score": 0.92, "index": 83}, {"type": "text", "coordinates": [471, 484, 541, 502], "content": ". Choose any", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [69, 499, 110, 515], "content": "integer ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [111, 502, 116, 511], "content": "\\ell", "score": 0.89, "index": 86}, {"type": "text", "coordinates": [117, 499, 179, 515], "content": "coprime to ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [180, 502, 198, 511], "content": "\\kappa N", "score": 0.91, "index": 88}, {"type": "text", "coordinates": [198, 499, 278, 515], "content": ". Then for any ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [279, 502, 316, 514], "content": "\\lambda\\in P_{+}", "score": 0.94, "index": 90}, {"type": "text", "coordinates": [316, 499, 449, 515], "content": " there is a unique weight ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [449, 500, 499, 514], "content": "\\lambda^{(\\ell)}\\in P_{+}", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [500, 499, 542, 515], "content": ", coroot", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [71, 520, 79, 525], "content": "\\alpha", "score": 0.85, "index": 94}, {"type": "text", "coordinates": [80, 514, 223, 529], "content": ", and (finite) Weyl element ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [223, 520, 232, 525], "content": "\\omega", "score": 0.89, "index": 96}, {"type": "text", "coordinates": [232, 514, 286, 529], "content": " such that", "score": 1.0, "index": 97}, {"type": "interline_equation", "coordinates": [228, 541, 383, 557], "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "score": 0.91, "index": 98}, {"type": "text", "coordinates": [70, 572, 369, 589], "content": "This is simply the statement that the affine Weyl orbit of ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [370, 573, 412, 586], "content": "\\ell\\left(\\lambda+\\rho\\right)", "score": 0.93, "index": 100}, {"type": "text", "coordinates": [412, 572, 505, 589], "content": " intersects the set ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [505, 575, 540, 586], "content": "P_{+}+\\rho", "score": 0.93, "index": 102}, {"type": "text", "coordinates": [70, 587, 234, 603], "content": "at precisely one point (namely ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [234, 587, 277, 600], "content": "\\lambda^{(\\ell)}+\\rho)", "score": 0.93, "index": 104}, {"type": "text", "coordinates": [277, 587, 320, 603], "content": ". Write ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [320, 587, 420, 601], "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname{det}\\omega=\\pm1", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [421, 587, 479, 603], "content": ". Then [16]", "score": 1.0, "index": 107}, {"type": "interline_equation", "coordinates": [236, 614, 374, 631], "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [71, 644, 541, 661], "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [71, 661, 81, 672], "content": "\\mathbb{Q}", "score": 0.88, "index": 110}, {"type": "text", "coordinates": [81, 659, 154, 675], "content": " by all entries ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [154, 661, 174, 673], "content": "S_{\\lambda\\mu}", "score": 0.92, "index": 112}, {"type": "text", "coordinates": [174, 659, 316, 675], "content": " lies in the cyclotomic field ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [316, 660, 357, 673], "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "score": 0.92, "index": 114}, {"type": "text", "coordinates": [358, 659, 395, 675], "content": " where ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [395, 661, 407, 672], "content": "\\xi_{n}", "score": 0.92, "index": 116}, {"type": "text", "coordinates": [407, 659, 540, 675], "content": " denotes the root of unity", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [71, 674, 127, 687], "content": "\\exp[2\\pi\\mathrm{i}/n]", "score": 0.92, "index": 118}, {"type": "text", "coordinates": [127, 671, 173, 692], "content": "; for any ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [173, 674, 327, 687], "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [327, 671, 458, 692], "content": ", there will be a function ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [459, 675, 540, 687], "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "score": 0.94, "index": 122}, {"type": "text", "coordinates": [71, 688, 122, 701], "content": "such that", "score": 1.0, "index": 123}, {"type": "interline_equation", "coordinates": [204, 701, 407, 718], "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "score": 0.91, "index": 124}]	[]	[{"type": "block", "coordinates": [250, 117, 361, 132], "content": "", "caption": ""}, {"type": "block", "coordinates": [235, 278, 376, 293], "content": "", "caption": ""}, {"type": "block", "coordinates": [201, 426, 409, 443], "content": "", "caption": ""}, {"type": "block", "coordinates": [228, 541, 383, 557], "content": "", "caption": ""}, {"type": "block", "coordinates": [236, 614, 374, 631], 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"\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Charge-conjugation is the order 2 permutation of $P_{+}$ given by $C\\lambda\\,=\\,^{t}\\lambda$ , the weight contragredient to $\\lambda$ . For instance $C0=0$ . It has the basic property that ", "page_idx": 3}, {"type": "equation", "text": "$$\nS_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and $S^{2}=C$ . $C$ corresponds to a symmetry of the (unextended) Dynkin diagram of $X_{r}$ , as we will see next section. ", "page_idx": 3}, {"type": "text", "text": "Related to $C$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $X_{r}^{(1)}$ with nontrivial conjugations other than chargeconjugation are $D_{e v e n}^{(1)}$ . ", "page_idx": 3}, {"type": "text", "text": "Another important symmetry of the matrix $S$ is called simple-currents. Any weight $j\\in P_{+}$ with q-dimension $\\mathcal{D}(j)=1$ , is called a simple-current. To any such weight $j$ is associated a permutation $J$ of $P_{+}$ and a function $Q_{j}:P_{+}\\to\\mathbb{Q}$ , such that $J0=j$ and ", "page_idx": 3}, {"type": "equation", "text": "$$\nS_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "The simple-currents form an abelian group, given by composition of the permutations $J$ . ", "page_idx": 3}, {"type": "text", "text": "All simple-currents for the affine algebras were classified in [12], and with one unimportant exception ( ${E}_{8}^{(1)}$ at level 2) correspond to symmetries of the extended Coxeter\u2013Dynkin diagram of $X_{r}^{(1)}$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $W_{0}^{+}$ . ", "page_idx": 3}, {"type": "text", "text": "Evaluating $S_{J\\lambda,j^{\\prime}}$ in two ways gives the useful ", "page_idx": 3}, {"type": "equation", "text": "$$\nQ_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and hence the reciprocity $Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)$ . ", "page_idx": 3}, {"type": "text", "text": "For each $X_{r}$ , the inner products $(\\lambda\|\\mu)$ of weights are rational; let $N$ denote the least common denominator. E.g. for $A_{r}$ this is $N=r+1$ , while for $E_{8}$ it is $N=1$ . Choose any integer $\\ell$ coprime to $\\kappa N$ . Then for any $\\lambda\\in P_{+}$ there is a unique weight $\\lambda^{(\\ell)}\\in P_{+}$ , coroot $\\alpha$ , and (finite) Weyl element $\\omega$ such that ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "This is simply the statement that the affine Weyl orbit of $\\ell\\left(\\lambda+\\rho\\right)$ intersects the set $P_{+}+\\rho$ at precisely one point (namely $\\lambda^{(\\ell)}+\\rho)$ . Write $\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname{det}\\omega=\\pm1$ . Then [16] ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over $\\mathbb{Q}$ by all entries $S_{\\lambda\\mu}$ lies in the cyclotomic field $\\mathbb{Q}[\\xi_{4N\\kappa}]$ where $\\xi_{n}$ denotes the root of unity $\\exp[2\\pi\\mathrm{i}/n]$ ; for any $\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}$ , there will be a function $\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}$ such that ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .\n$$", 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", "type": "text"}, {"bbox": [146, 148, 155, 157], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [156, 145, 520, 162], "score": 1.0, "content": " corresponds to a symmetry of the (unextended) Dynkin diagram of ", "type": "text"}, {"bbox": [520, 148, 536, 159], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [537, 145, 540, 162], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [71, 161, 212, 174], "spans": [{"bbox": [71, 161, 212, 174], "score": 1.0, "content": "as we will see next section.", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [70, 173, 541, 219], "lines": [{"bbox": [94, 174, 540, 190], "spans": [{"bbox": [94, 174, 155, 190], "score": 1.0, "content": "Related to ", "type": "text"}, {"bbox": [155, 177, 165, 185], "score": 0.9, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [165, 174, 540, 190], "score": 1.0, "content": " are all the other symmetries of the unextended Dynkin diagram. We", "type": "text"}], "index": 5}, {"bbox": [68, 186, 542, 209], "spans": [{"bbox": [68, 186, 256, 209], "score": 1.0, "content": "call these conjugations. 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Any weight", "type": "text"}], "index": 8}, {"bbox": [71, 236, 541, 250], "spans": [{"bbox": [71, 238, 110, 249], "score": 0.93, "content": "j\\in P_{+}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [110, 236, 210, 250], "score": 1.0, "content": " with q-dimension ", "type": "text"}, {"bbox": [210, 237, 260, 249], "score": 0.94, "content": "\\mathcal{D}(j)=1", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [261, 236, 520, 250], "score": 1.0, "content": ", is called a simple-current. 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The simplest proof would use the methods of Proposition 4.1 below. For", "type": "text"}], "index": 15}, {"bbox": [70, 371, 540, 385], "spans": [{"bbox": [70, 371, 540, 385], "score": 1.0, "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "type": "text"}], "index": 16}, {"bbox": [69, 384, 188, 402], "spans": [{"bbox": [69, 384, 162, 402], "score": 1.0, "content": "group is denoted ", "type": "text"}, {"bbox": [162, 385, 183, 398], "score": 0.93, "content": "W_{0}^{+}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [184, 384, 188, 402], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15}, {"type": "text", "bbox": [94, 397, 338, 412], "lines": [{"bbox": [95, 399, 336, 414], "spans": [{"bbox": [95, 399, 155, 414], "score": 1.0, "content": "Evaluating ", "type": "text"}, {"bbox": [155, 401, 185, 413], "score": 0.93, "content": "S_{J\\lambda,j^{\\prime}}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [186, 399, 336, 414], "score": 1.0, "content": " in two ways gives the useful", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 455, 294, 469], "lines": [{"bbox": [70, 457, 293, 472], "spans": [{"bbox": [70, 457, 208, 472], "score": 1.0, "content": "and hence the reciprocity ", "type": "text"}, {"bbox": [208, 458, 289, 471], "score": 0.93, "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [290, 457, 293, 472], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 470, 541, 527], "lines": [{"bbox": [94, 471, 541, 487], "spans": [{"bbox": [94, 471, 143, 487], "score": 1.0, "content": "For each ", "type": "text"}, {"bbox": [144, 473, 159, 484], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [159, 471, 267, 487], "score": 1.0, "content": ", the inner products ", "type": "text"}, {"bbox": [268, 473, 295, 485], "score": 0.93, "content": "(\\lambda\|\\mu)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [295, 471, 441, 487], "score": 1.0, "content": " of weights are rational; let ", "type": "text"}, {"bbox": [441, 473, 452, 482], "score": 0.89, "content": "N", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [452, 471, 541, 487], "score": 1.0, "content": " denote the least", "type": "text"}], "index": 21}, {"bbox": [69, 484, 541, 502], "spans": [{"bbox": [69, 484, 235, 502], "score": 1.0, "content": "common denominator. E.g. for ", "type": "text"}, {"bbox": [235, 488, 250, 498], "score": 0.92, "content": "A_{r}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [250, 484, 288, 502], "score": 1.0, "content": " this is ", "type": "text"}, {"bbox": [288, 488, 341, 497], "score": 0.9, "content": "N=r+1", "type": "inline_equation", "height": 9, "width": 53}, {"bbox": [341, 484, 396, 502], "score": 1.0, "content": ", while for ", "type": "text"}, {"bbox": [396, 487, 411, 498], "score": 0.86, "content": "E_{8}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [411, 484, 437, 502], "score": 1.0, "content": " it is ", "type": "text"}, {"bbox": [438, 488, 471, 497], "score": 0.92, "content": "N=1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [471, 484, 541, 502], "score": 1.0, "content": ". Choose any", "type": "text"}], "index": 22}, {"bbox": [69, 499, 542, 515], "spans": [{"bbox": [69, 499, 110, 515], "score": 1.0, "content": "integer ", "type": "text"}, {"bbox": [111, 502, 116, 511], "score": 0.89, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [117, 499, 179, 515], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [180, 502, 198, 511], "score": 0.91, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [198, 499, 278, 515], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [279, 502, 316, 514], "score": 0.94, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [316, 499, 449, 515], "score": 1.0, "content": " there is a unique weight ", "type": "text"}, {"bbox": [449, 500, 499, 514], "score": 0.93, "content": "\\lambda^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [500, 499, 542, 515], "score": 1.0, "content": ", coroot", "type": "text"}], "index": 23}, {"bbox": [71, 514, 286, 529], "spans": [{"bbox": [71, 520, 79, 525], "score": 0.85, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [80, 514, 223, 529], "score": 1.0, "content": ", and (finite) Weyl element ", "type": "text"}, {"bbox": [223, 520, 232, 525], "score": 0.89, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [232, 514, 286, 529], "score": 1.0, "content": " such that", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [70, 570, 541, 600], "lines": [{"bbox": [70, 572, 540, 589], "spans": [{"bbox": [70, 572, 369, 589], "score": 1.0, "content": "This is simply the statement that the affine Weyl orbit of ", "type": "text"}, {"bbox": [370, 573, 412, 586], "score": 0.93, "content": "\\ell\\left(\\lambda+\\rho\\right)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [412, 572, 505, 589], "score": 1.0, "content": " intersects the set ", "type": "text"}, {"bbox": [505, 575, 540, 586], "score": 0.93, "content": "P_{+}+\\rho", "type": "inline_equation", "height": 11, "width": 35}], "index": 26}, {"bbox": [70, 587, 479, 603], "spans": [{"bbox": [70, 587, 234, 603], "score": 1.0, "content": "at precisely one point (namely ", "type": "text"}, {"bbox": [234, 587, 277, 600], "score": 0.93, "content": "\\lambda^{(\\ell)}+\\rho)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [277, 587, 320, 603], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [320, 587, 420, 601], "score": 0.92, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname{det}\\omega=\\pm1", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [421, 587, 479, 603], "score": 1.0, "content": ". Then [16]", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [68, 642, 541, 699], "lines": [{"bbox": [71, 644, 541, 661], "spans": [{"bbox": [71, 644, 541, 661], "score": 1.0, "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "type": "text"}], "index": 29}, {"bbox": [71, 659, 540, 675], "spans": [{"bbox": [71, 661, 81, 672], "score": 0.88, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [81, 659, 154, 675], "score": 1.0, "content": " by all entries ", "type": "text"}, {"bbox": [154, 661, 174, 673], "score": 0.92, "content": "S_{\\lambda\\mu}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [174, 659, 316, 675], "score": 1.0, "content": " lies in the cyclotomic field ", "type": "text"}, {"bbox": [316, 660, 357, 673], "score": 0.92, "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [358, 659, 395, 675], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [395, 661, 407, 672], "score": 0.92, "content": "\\xi_{n}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [407, 659, 540, 675], "score": 1.0, "content": " denotes the root of unity", "type": "text"}], "index": 30}, {"bbox": [71, 671, 540, 692], "spans": [{"bbox": [71, 674, 127, 687], "score": 0.92, "content": "\\exp[2\\pi\\mathrm{i}/n]", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [127, 671, 173, 692], "score": 1.0, "content": "; for any ", "type": "text"}, {"bbox": [173, 674, 327, 687], "score": 0.91, "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "type": "inline_equation", "height": 13, "width": 154}, {"bbox": [327, 671, 458, 692], "score": 1.0, "content": ", there will be a function ", "type": "text"}, {"bbox": [459, 675, 540, 687], "score": 0.94, "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "type": "inline_equation", "height": 12, "width": 81}], "index": 31}, {"bbox": [71, 688, 122, 701], "spans": [{"bbox": [71, 688, 122, 701], "score": 1.0, "content": "such that", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [250, 117, 361, 132], "lines": [{"bbox": [250, 117, 361, 132], "spans": [{"bbox": [250, 117, 361, 132], "score": 0.93, "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "lines": [{"bbox": [235, 278, 376, 293], "spans": [{"bbox": [235, 278, 376, 293], "score": 0.93, "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 540, 100], "lines": [{"bbox": [95, 72, 542, 90], "spans": [{"bbox": [95, 72, 360, 90], "score": 1.0, "content": "Charge-conjugation is the order 2 permutation of ", "type": "text"}, {"bbox": [360, 75, 376, 87], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [376, 72, 429, 90], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [430, 74, 476, 84], "score": 0.93, "content": "C\\lambda\\,=\\,^{t}\\lambda", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [477, 72, 542, 90], "score": 1.0, "content": ", the weight", "type": "text"}], "index": 0}, {"bbox": [70, 88, 452, 104], "spans": [{"bbox": [70, 88, 165, 104], "score": 1.0, "content": "contragredient to ", "type": "text"}, {"bbox": [165, 90, 172, 99], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [173, 88, 248, 104], "score": 1.0, "content": ". For instance ", "type": "text"}, {"bbox": [248, 90, 286, 99], "score": 0.93, "content": "C0=0", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 88, 452, 104], "score": 1.0, "content": ". It has the basic property that", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [70, 72, 542, 104]}, {"type": "interline_equation", "bbox": [250, 117, 361, 132], "lines": [{"bbox": [250, 117, 361, 132], "spans": [{"bbox": [250, 117, 361, 132], "score": 0.93, "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{}", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 143, 540, 172], "lines": [{"bbox": [70, 145, 540, 162], "spans": [{"bbox": [70, 145, 95, 162], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 146, 136, 156], "score": 0.92, "content": "S^{2}=C", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [136, 145, 145, 162], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [146, 148, 155, 157], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [156, 145, 520, 162], "score": 1.0, "content": " corresponds to a symmetry of the (unextended) Dynkin diagram of ", "type": "text"}, {"bbox": [520, 148, 536, 159], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [537, 145, 540, 162], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [71, 161, 212, 174], "spans": [{"bbox": [71, 161, 212, 174], "score": 1.0, "content": "as we will see next section.", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [70, 145, 540, 174]}, {"type": "text", "bbox": [70, 173, 541, 219], "lines": [{"bbox": [94, 174, 540, 190], "spans": [{"bbox": [94, 174, 155, 190], "score": 1.0, "content": "Related to ", "type": "text"}, {"bbox": [155, 177, 165, 185], "score": 0.9, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [165, 174, 540, 190], "score": 1.0, "content": " are all the other symmetries of the unextended Dynkin diagram. We", "type": "text"}], "index": 5}, {"bbox": [68, 186, 542, 209], "spans": [{"bbox": [68, 186, 256, 209], "score": 1.0, "content": "call these conjugations. The only ", "type": "text"}, {"bbox": [256, 188, 280, 203], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [281, 186, 542, 209], "score": 1.0, "content": "with nontrivial conjugations other than charge-", "type": "text"}], "index": 6}, {"bbox": [70, 203, 192, 226], "spans": [{"bbox": [70, 203, 156, 226], "score": 1.0, "content": "conjugation are ", "type": "text"}, {"bbox": [156, 204, 187, 219], "score": 0.92, "content": "D_{e v e n}^{(1)}", "type": "inline_equation", "height": 15, "width": 31}, {"bbox": [187, 203, 192, 226], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [68, 174, 542, 226]}, {"type": "text", "bbox": [70, 219, 541, 263], "lines": [{"bbox": [95, 221, 541, 236], "spans": [{"bbox": [95, 221, 330, 236], "score": 1.0, "content": "Another important symmetry of the matrix ", "type": "text"}, {"bbox": [330, 223, 339, 232], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 221, 541, 236], "score": 1.0, "content": " is called simple-currents. Any weight", "type": "text"}], "index": 8}, {"bbox": [71, 236, 541, 250], "spans": [{"bbox": [71, 238, 110, 249], "score": 0.93, "content": "j\\in P_{+}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [110, 236, 210, 250], "score": 1.0, "content": " with q-dimension ", "type": "text"}, {"bbox": [210, 237, 260, 249], "score": 0.94, "content": "\\mathcal{D}(j)=1", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [261, 236, 520, 250], "score": 1.0, "content": ", is called a simple-current. To any such weight ", "type": "text"}, {"bbox": [520, 238, 527, 249], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [527, 236, 541, 250], "score": 1.0, "content": " is", "type": "text"}], "index": 9}, {"bbox": [71, 250, 520, 264], "spans": [{"bbox": [71, 250, 206, 264], "score": 1.0, "content": "associated a permutation ", "type": "text"}, {"bbox": [207, 252, 214, 261], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [215, 250, 231, 264], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [232, 252, 248, 263], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 250, 331, 264], "score": 1.0, "content": " and a function ", "type": "text"}, {"bbox": [332, 252, 400, 264], "score": 0.93, "content": "Q_{j}:P_{+}\\to\\mathbb{Q}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [401, 250, 459, 264], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [460, 252, 495, 263], "score": 0.94, "content": "J0=j", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [496, 250, 520, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 10}], "index": 9, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [71, 221, 541, 264]}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "lines": [{"bbox": [235, 278, 376, 293], "spans": [{"bbox": [235, 278, 376, 293], "score": 0.93, "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 306, 538, 320], "lines": [{"bbox": [71, 308, 537, 323], "spans": [{"bbox": [71, 308, 525, 323], "score": 1.0, "content": "The simple-currents form an abelian group, given by composition of the permutations ", "type": "text"}, {"bbox": [525, 310, 533, 319], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [533, 308, 537, 323], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [71, 308, 537, 323]}, {"type": "text", "bbox": [70, 321, 541, 397], "lines": [{"bbox": [94, 322, 541, 338], "spans": [{"bbox": [94, 322, 541, 338], "score": 1.0, "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-", "type": "text"}], "index": 13}, {"bbox": [69, 334, 541, 358], "spans": [{"bbox": [69, 334, 154, 358], "score": 1.0, "content": "tant exception (", "type": "text"}, {"bbox": [154, 336, 176, 353], "score": 0.92, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [177, 334, 541, 358], "score": 1.0, "content": "at level 2) correspond to symmetries of the extended Coxeter\u2013Dynkin", "type": "text"}], "index": 14}, {"bbox": [69, 353, 542, 372], "spans": [{"bbox": [69, 353, 130, 372], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [131, 354, 154, 368], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [155, 353, 542, 372], "score": 1.0, "content": ". The simplest proof would use the methods of Proposition 4.1 below. For", "type": "text"}], "index": 15}, {"bbox": [70, 371, 540, 385], "spans": [{"bbox": [70, 371, 540, 385], "score": 1.0, "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "type": "text"}], "index": 16}, {"bbox": [69, 384, 188, 402], "spans": [{"bbox": [69, 384, 162, 402], "score": 1.0, "content": "group is denoted ", "type": "text"}, {"bbox": [162, 385, 183, 398], "score": 0.93, "content": "W_{0}^{+}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [184, 384, 188, 402], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [69, 322, 542, 402]}, {"type": "text", "bbox": [94, 397, 338, 412], "lines": [{"bbox": [95, 399, 336, 414], "spans": [{"bbox": [95, 399, 155, 414], "score": 1.0, "content": "Evaluating ", "type": "text"}, {"bbox": [155, 401, 185, 413], "score": 0.93, "content": "S_{J\\lambda,j^{\\prime}}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [186, 399, 336, 414], "score": 1.0, "content": " in two ways gives the useful", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [95, 399, 336, 414]}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 455, 294, 469], "lines": [{"bbox": [70, 457, 293, 472], "spans": [{"bbox": [70, 457, 208, 472], "score": 1.0, "content": "and hence the reciprocity ", "type": "text"}, {"bbox": [208, 458, 289, 471], "score": 0.93, "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [290, 457, 293, 472], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [70, 457, 293, 472]}, {"type": "text", "bbox": [70, 470, 541, 527], "lines": [{"bbox": [94, 471, 541, 487], "spans": [{"bbox": [94, 471, 143, 487], "score": 1.0, "content": "For each ", "type": "text"}, {"bbox": [144, 473, 159, 484], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [159, 471, 267, 487], "score": 1.0, "content": ", the inner products ", "type": "text"}, {"bbox": [268, 473, 295, 485], "score": 0.93, "content": "(\\lambda\|\\mu)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [295, 471, 441, 487], "score": 1.0, "content": " of weights are rational; let ", "type": "text"}, {"bbox": [441, 473, 452, 482], "score": 0.89, "content": "N", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [452, 471, 541, 487], "score": 1.0, "content": " denote the least", "type": "text"}], "index": 21}, {"bbox": [69, 484, 541, 502], "spans": [{"bbox": [69, 484, 235, 502], "score": 1.0, "content": "common denominator. E.g. for ", "type": "text"}, {"bbox": [235, 488, 250, 498], "score": 0.92, "content": "A_{r}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [250, 484, 288, 502], "score": 1.0, "content": " this is ", "type": "text"}, {"bbox": [288, 488, 341, 497], "score": 0.9, "content": "N=r+1", "type": "inline_equation", "height": 9, "width": 53}, {"bbox": [341, 484, 396, 502], "score": 1.0, "content": ", while for ", "type": "text"}, {"bbox": [396, 487, 411, 498], "score": 0.86, "content": "E_{8}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [411, 484, 437, 502], "score": 1.0, "content": " it is ", "type": "text"}, {"bbox": [438, 488, 471, 497], "score": 0.92, "content": "N=1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [471, 484, 541, 502], "score": 1.0, "content": ". Choose any", "type": "text"}], "index": 22}, {"bbox": [69, 499, 542, 515], "spans": [{"bbox": [69, 499, 110, 515], "score": 1.0, "content": "integer ", "type": "text"}, {"bbox": [111, 502, 116, 511], "score": 0.89, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [117, 499, 179, 515], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [180, 502, 198, 511], "score": 0.91, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [198, 499, 278, 515], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [279, 502, 316, 514], "score": 0.94, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [316, 499, 449, 515], "score": 1.0, "content": " there is a unique weight ", "type": "text"}, {"bbox": [449, 500, 499, 514], "score": 0.93, "content": "\\lambda^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [500, 499, 542, 515], "score": 1.0, "content": ", coroot", "type": "text"}], "index": 23}, {"bbox": [71, 514, 286, 529], "spans": [{"bbox": [71, 520, 79, 525], "score": 0.85, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [80, 514, 223, 529], "score": 1.0, "content": ", and (finite) Weyl element ", "type": "text"}, {"bbox": [223, 520, 232, 525], "score": 0.89, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [232, 514, 286, 529], "score": 1.0, "content": " such that", "type": "text"}], "index": 24}], "index": 22.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [69, 471, 542, 529]}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 570, 541, 600], "lines": [{"bbox": [70, 572, 540, 589], "spans": [{"bbox": [70, 572, 369, 589], "score": 1.0, "content": "This is simply the statement that the affine Weyl orbit of ", "type": "text"}, {"bbox": [370, 573, 412, 586], "score": 0.93, "content": "\\ell\\left(\\lambda+\\rho\\right)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [412, 572, 505, 589], "score": 1.0, "content": " intersects the set ", "type": "text"}, {"bbox": [505, 575, 540, 586], "score": 0.93, "content": "P_{+}+\\rho", "type": "inline_equation", "height": 11, "width": 35}], "index": 26}, {"bbox": [70, 587, 479, 603], "spans": [{"bbox": [70, 587, 234, 603], "score": 1.0, "content": "at precisely one point (namely ", "type": "text"}, {"bbox": [234, 587, 277, 600], "score": 0.93, "content": "\\lambda^{(\\ell)}+\\rho)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [277, 587, 320, 603], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [320, 587, 420, 601], "score": 0.92, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [421, 587, 479, 603], "score": 1.0, "content": ". Then [16]", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [70, 572, 540, 603]}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [68, 642, 541, 699], "lines": [{"bbox": [71, 644, 541, 661], "spans": [{"bbox": [71, 644, 541, 661], "score": 1.0, "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "type": "text"}], "index": 29}, {"bbox": [71, 659, 540, 675], "spans": [{"bbox": [71, 661, 81, 672], "score": 0.88, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [81, 659, 154, 675], "score": 1.0, "content": " by all entries ", "type": "text"}, {"bbox": [154, 661, 174, 673], "score": 0.92, "content": "S_{\\lambda\\mu}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [174, 659, 316, 675], "score": 1.0, "content": " lies in the cyclotomic field ", "type": "text"}, {"bbox": [316, 660, 357, 673], "score": 0.92, "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [358, 659, 395, 675], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [395, 661, 407, 672], "score": 0.92, "content": "\\xi_{n}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [407, 659, 540, 675], "score": 1.0, "content": " denotes the root of unity", "type": "text"}], "index": 30}, {"bbox": [71, 671, 540, 692], "spans": [{"bbox": [71, 674, 127, 687], "score": 0.92, "content": "\\exp[2\\pi\\mathrm{i}/n]", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [127, 671, 173, 692], "score": 1.0, "content": "; for any ", "type": "text"}, {"bbox": [173, 674, 327, 687], "score": 0.91, "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "type": "inline_equation", "height": 13, "width": 154}, {"bbox": [327, 671, 458, 692], "score": 1.0, "content": ", there will be a function ", "type": "text"}, {"bbox": [459, 675, 540, 687], "score": 0.94, "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "type": "inline_equation", "height": 12, "width": 81}], "index": 31}, {"bbox": [71, 688, 122, 701], "spans": [{"bbox": [71, 688, 122, 701], "score": 1.0, "content": "such that", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [71, 644, 541, 701]}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33, "page_num": "page_3", "page_size": [612.0, 792.0]}]}
0002044v1	10	whose diagram is the transpose of that for $$\lambda$$ . (For this purpose the algebra $$C_{1}$$ may be identified with $$A_{1}$$ .) For example, $$\tau\Lambda_{a}=a\tilde{\Lambda}_{1}$$ , where we use tilde’s to denote the quantities of $$C_{k,r}$$ . In fact, $$\tau:P_{+}(C_{r,k})\rightarrow P_{+}(C_{k,r})$$ is a bijection. Then This rank-level duality for $$C_{r}^{(1)}$$ is especially interesting, as it defines a fusion ring iso- morphism $$\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$$ (see §5). When $$k=r$$ , we get a nontrivial fusion-symmetry: $$\pi_{\mathrm{rld}}\lambda\,{\overset{\mathrm{def}}{=}}\,\tau\lambda$$ . The only fusion product we need is valid for $$i<r$$ and $$k\geq2$$ . The following character formula (2.1b) will also be used: where $$\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$$ as before. Theorem 3.C. The fusion-symmetries for $$C_{r}^{(1)}$$ level $$k$$ , when $$k\neq r$$ and either $$k$$ or $$r$$ is even, are $$\pi[1]^{i}$$ for $$i\in\{0,1\}$$ . When $$k\neq r$$ but both $$k$$ and $$r$$ are odd, then there is no nontrivial fusion-symmetry. When $$k=r$$ , they are $$\pi[1]^{i}\,\pi_{\mathrm{rld}}^{j}$$ $$\mathit{\Pi}_{k}$$ even) or $$\pi[1]^{i}$$ ( $$\mathit{k}$$ odd), for $$i,j\in\{0,1\}$$ . When $$r=k$$ is even, $$A(C_{r,k})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$ . 3.4. The algebra $$D_{r}^{(1)},\,r\geq4$$ A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!\cdot\!\cdot\!+\!2\lambda_{r-2}\!+\!\lambda_{r-1}\!+\!\lambda_{r}$$ , and $$\kappa=k{+}2r{-}2$$ . For any $$r$$ , there are the conjugations $$C_{0}=i d$$ . and $$C_{1}\lambda=(\lambda_{0},\lambda_{1},.\dots,\lambda_{r-2},\lambda_{r},\lambda_{r-1})$$ . The charge-conjugation $$C$$ equals $$C_{1}$$ for odd $$r$$ , and $$C_{0}$$ for even $$r$$ . When $$r=4$$ there are four additional conjugations; these six $$C_{i}$$ correspond to all permutations of the $${D}_{4}^{(1)}$$ Dynkin labels $$\lambda_{1},\lambda_{3},\lambda_{4}$$ . There are three non-trivial simple-currents, $$J_{v}$$ , $$J_{s}$$ and $$J_{v}J_{s}$$ . Explicitly, we have $$J_{v}\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},...\,,\lambda_{r-2},\lambda_{r},\lambda_{r-1}\right)$$ with $$Q_{v}(\lambda)=(\lambda_{r-1}+\lambda_{r})/2$$ , and with $$\begin{array}{r}{Q_{s}(\lambda)=(2\sum_{j=1}^{r-2}j\lambda_{j}\!-\!(r\!-\!2)\lambda_{r-1}\!-\!r\lambda_{r})/4}\end{array}$$ . From this we compute $$Q_{s}(J_{s}0)=-r k/4$$ . The fusion products we need are	<p>whose diagram is the transpose of that for $$\lambda$$ . (For this purpose the algebra $$C_{1}$$ may be identified with $$A_{1}$$ .) For example, $$\tau\Lambda_{a}=a\tilde{\Lambda}_{1}$$ , where we use tilde’s to denote the quantities of $$C_{k,r}$$ . In fact, $$\tau:P_{+}(C_{r,k})\rightarrow P_{+}(C_{k,r})$$ is a bijection. Then</p> <p>This rank-level duality for $$C_{r}^{(1)}$$ is especially interesting, as it defines a fusion ring iso- morphism $$\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$$ (see §5). When $$k=r$$ , we get a nontrivial fusion-symmetry: $$\pi_{\mathrm{rld}}\lambda\,{\overset{\mathrm{def}}{=}}\,\tau\lambda$$ .</p> <p>The only fusion product we need is</p> <p>valid for $$i<r$$ and $$k\geq2$$ . The following character formula (2.1b) will also be used:</p> <p>where $$\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$$ as before.</p> <p>Theorem 3.C. The fusion-symmetries for $$C_{r}^{(1)}$$ level $$k$$ , when $$k\neq r$$ and either $$k$$ or $$r$$ is even, are $$\pi[1]^{i}$$ for $$i\in\{0,1\}$$ . When $$k\neq r$$ but both $$k$$ and $$r$$ are odd, then there is no nontrivial fusion-symmetry. When $$k=r$$ , they are $$\pi[1]^{i}\,\pi_{\mathrm{rld}}^{j}$$ $$\mathit{\Pi}_{k}$$ even) or $$\pi[1]^{i}$$ ( $$\mathit{k}$$ odd), for $$i,j\in\{0,1\}$$ .</p> <p>When $$r=k$$ is even, $$A(C_{r,k})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$ .</p> <p>3.4. The algebra $$D_{r}^{(1)},\,r\geq4$$</p> <p>A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!\cdot\!\cdot\!+\!2\lambda_{r-2}\!+\!\lambda_{r-1}\!+\!\lambda_{r}$$ , and $$\kappa=k{+}2r{-}2$$ . For any $$r$$ , there are the conjugations $$C_{0}=i d$$ . and $$C_{1}\lambda=(\lambda_{0},\lambda_{1},.\dots,\lambda_{r-2},\lambda_{r},\lambda_{r-1})$$ . The charge-conjugation $$C$$ equals $$C_{1}$$ for odd $$r$$ , and $$C_{0}$$ for even $$r$$ . When $$r=4$$ there are four additional conjugations; these six $$C_{i}$$ correspond to all permutations of the $${D}_{4}^{(1)}$$ Dynkin labels $$\lambda_{1},\lambda_{3},\lambda_{4}$$ .</p> <p>There are three non-trivial simple-currents, $$J_{v}$$ , $$J_{s}$$ and $$J_{v}J_{s}$$ . Explicitly, we have $$J_{v}\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},...\,,\lambda_{r-2},\lambda_{r},\lambda_{r-1}\right)$$ with $$Q_{v}(\lambda)=(\lambda_{r-1}+\lambda_{r})/2$$ , and</p> <p>with $$\begin{array}{r}{Q_{s}(\lambda)=(2\sum_{j=1}^{r-2}j\lambda_{j}\!-\!(r\!-\!2)\lambda_{r-1}\!-\!r\lambda_{r})/4}\end{array}$$ . From this we compute $$Q_{s}(J_{s}0)=-r k/4$$ . The fusion products we need are</p>	[{"type": "text", "coordinates": [70, 70, 542, 115], "content": "whose diagram is the transpose of that for $$\\lambda$$ . (For this purpose the algebra $$C_{1}$$ may be\nidentified with $$A_{1}$$ .) For example, $$\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}$$ , where we use tilde\u2019s to denote the quantities\nof $$C_{k,r}$$ . In fact, $$\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})$$ is a bijection. Then", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [267, 129, 344, 146], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 158, 541, 205], "content": "This rank-level duality for $$C_{r}^{(1)}$$ is especially interesting, as it defines a fusion ring iso-\nmorphism $$\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$$ (see \u00a75). When $$k=r$$ , we get a nontrivial fusion-symmetry:\n$$\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [95, 207, 282, 221], "content": "The only fusion product we need is", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [201, 234, 406, 251], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [71, 262, 505, 277], "content": "valid for $$i<r$$ and $$k\\geq2$$ . The following character formula (2.1b) will also be used:", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [209, 289, 401, 329], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 339, 258, 355], "content": "where $$\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$$ as before.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 361, 542, 421], "content": "Theorem 3.C. The fusion-symmetries for $$C_{r}^{(1)}$$ level $$k$$ , when $$k\\neq r$$ and either $$k$$ or\n$$r$$ is even, are $$\\pi[1]^{i}$$ for $$i\\in\\{0,1\\}$$ . When $$k\\neq r$$ but both $$k$$ and $$r$$ are odd, then there is no\nnontrivial fusion-symmetry. When $$k=r$$ , they are $$\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}$$ $$\\mathit{\\Pi}_{k}$$ even) or $$\\pi[1]^{i}$$ ( $$\\mathit{k}$$ odd), for\n$$i,j\\in\\{0,1\\}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [94, 427, 306, 443], "content": "When $$r=k$$ is even, $$A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [71, 456, 221, 474], "content": "3.4. The algebra $$D_{r}^{(1)},\\,r\\geq4$$", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 479, 541, 554], "content": "A weight $$\\lambda$$ of $$P_{+}$$ satisfies $$k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}$$ , and $$\\kappa=k{+}2r{-}2$$ .\nFor any $$r$$ , there are the conjugations $$C_{0}=i d$$ . and $$C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})$$ . The\ncharge-conjugation $$C$$ equals $$C_{1}$$ for odd $$r$$ , and $$C_{0}$$ for even $$r$$ . When $$r=4$$ there are four\nadditional conjugations; these six $$C_{i}$$ correspond to all permutations of the $${D}_{4}^{(1)}$$ Dynkin\nlabels $$\\lambda_{1},\\lambda_{3},\\lambda_{4}$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [70, 554, 541, 584], "content": "There are three non-trivial simple-currents, $$J_{v}$$ , $$J_{s}$$ and $$J_{v}J_{s}$$ . Explicitly, we have\n$$J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)$$ with $$Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2$$ , and", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [174, 596, 430, 627], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [69, 639, 541, 669], "content": "with $$\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}$$ . From this we compute $$Q_{s}(J_{s}0)=-r k/4$$ .\nThe fusion products we need are", "block_type": "text", "index": 15}]	[{"type": "text", "coordinates": [72, 74, 303, 88], "content": "whose diagram is the transpose of that for ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [303, 75, 311, 84], "content": "\\lambda", "score": 0.88, "index": 2}, {"type": "text", "coordinates": [311, 74, 481, 88], "content": ". (For this purpose the algebra ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [482, 75, 495, 86], "content": "C_{1}", "score": 0.92, "index": 4}, {"type": "text", "coordinates": [496, 74, 541, 88], "content": " may be", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 87, 149, 103], "content": "identified with ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [149, 90, 164, 101], "content": "A_{1}", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [164, 87, 246, 103], "content": ".) For example,", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [247, 87, 304, 101], "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [304, 87, 541, 103], "content": ", where we use tilde\u2019s to denote the quantities", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [70, 102, 84, 117], "content": "of ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [85, 104, 107, 117], "content": "C_{k,r}", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [107, 102, 157, 117], "content": ". In fact, ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [157, 103, 286, 117], "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [286, 102, 394, 117], "content": " is a bijection. Then", "score": 1.0, "index": 15}, {"type": "interline_equation", "coordinates": [267, 129, 344, 146], "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "score": 0.91, "index": 16}, {"type": "text", "coordinates": [93, 159, 234, 178], "content": "This rank-level duality for ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [234, 158, 257, 174], "content": "C_{r}^{(1)}", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [258, 159, 540, 178], "content": "is especially interesting, as it defines a fusion ring iso-", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [71, 176, 127, 191], "content": "morphism ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [127, 176, 225, 190], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "score": 0.91, "index": 21}, {"type": "text", "coordinates": [226, 176, 311, 191], "content": " (see \u00a75). When ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [312, 176, 341, 188], "content": "k=r", "score": 0.89, "index": 23}, {"type": "text", "coordinates": [342, 176, 540, 191], "content": ", we get a nontrivial fusion-symmetry:", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [71, 192, 128, 208], "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [129, 189, 134, 210], "content": ".", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [95, 209, 281, 223], "content": "The only fusion product we need is", "score": 1.0, "index": 27}, {"type": "interline_equation", "coordinates": [201, 234, 406, 251], "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "score": 0.72, "index": 28}, {"type": "text", "coordinates": [70, 263, 118, 281], "content": "valid for ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [118, 267, 144, 276], "content": "i<r", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [145, 263, 171, 281], "content": " and ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [171, 264, 200, 277], "content": "k\\geq2", "score": 0.9, "index": 32}, {"type": "text", "coordinates": [201, 263, 506, 281], "content": ". The following character formula (2.1b) will also be used:", "score": 1.0, "index": 33}, {"type": "interline_equation", "coordinates": [209, 289, 401, 329], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [71, 341, 105, 357], "content": "where ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [105, 342, 203, 356], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [203, 341, 257, 357], "content": " as before.", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [93, 362, 324, 381], "content": "Theorem 3.C. The fusion-symmetries for ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [325, 362, 348, 378], "content": "C_{r}^{(1)}", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [348, 362, 378, 381], "content": "level ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [378, 365, 387, 377], "content": "k", "score": 0.75, "index": 41}, {"type": "text", "coordinates": [387, 362, 424, 381], "content": ", when ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [424, 365, 455, 379], "content": "k\\neq r", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [456, 362, 516, 381], "content": " and either ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [516, 366, 524, 377], "content": "k", "score": 0.69, "index": 45}, {"type": "text", "coordinates": [525, 362, 542, 381], "content": " or", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [71, 385, 78, 391], "content": "r", "score": 0.66, "index": 47}, {"type": "text", "coordinates": [78, 380, 146, 394], "content": " is even, are ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [146, 379, 171, 393], "content": "\\pi[1]^{i}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [171, 380, 193, 394], "content": " for ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [193, 379, 243, 394], "content": "i\\in\\{0,1\\}", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [243, 380, 286, 394], "content": ". When ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [286, 380, 317, 393], "content": "k\\neq r", "score": 0.91, "index": 53}, {"type": "text", "coordinates": [317, 380, 365, 394], "content": " but both ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [365, 380, 373, 391], "content": "k", "score": 0.77, "index": 55}, {"type": "text", "coordinates": [374, 380, 399, 394], "content": " and ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [399, 382, 407, 391], "content": "r", "score": 0.69, "index": 57}, {"type": "text", "coordinates": [407, 380, 541, 394], "content": " are odd, then there is no", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [70, 394, 255, 410], "content": "nontrivial fusion-symmetry. When ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [255, 394, 285, 406], "content": "k=r", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [285, 394, 337, 410], "content": ", they are ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [337, 392, 384, 409], "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "score": 0.91, "index": 62}, {"type": "text", "coordinates": [384, 394, 390, 410], "content": " ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [391, 393, 399, 406], "content": "\\mathit{\\Pi}_{k}", "score": 0.6, "index": 64}, {"type": "text", "coordinates": [400, 394, 449, 410], "content": " even) or ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [450, 393, 475, 408], "content": "\\pi[1]^{i}", "score": 0.79, "index": 66}, {"type": "text", "coordinates": [475, 394, 482, 410], "content": " (", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [483, 394, 491, 406], "content": "\\mathit{k}", "score": 0.43, "index": 68}, {"type": "text", "coordinates": [491, 394, 542, 410], "content": " odd), for", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [71, 409, 131, 422], "content": "i,j\\in\\{0,1\\}", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [131, 407, 136, 423], "content": ".", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [95, 429, 129, 444], "content": "When ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [130, 431, 159, 441], "content": "r=k", "score": 0.88, "index": 73}, {"type": "text", "coordinates": [159, 429, 204, 444], "content": " is even, ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [205, 429, 303, 444], "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [303, 429, 305, 444], "content": ".", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [70, 456, 160, 476], "content": "3.4. The algebra ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [161, 458, 219, 474], "content": "D_{r}^{(1)},\\,r\\geq4", "score": 0.3, "index": 78}, {"type": "text", "coordinates": [94, 481, 144, 497], "content": "A weight ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [144, 483, 151, 493], "content": "\\lambda", "score": 0.81, "index": 80}, {"type": "text", "coordinates": [151, 481, 166, 497], "content": " of ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [166, 484, 182, 496], "content": "P_{+}", "score": 0.89, "index": 82}, {"type": "text", "coordinates": [182, 481, 227, 497], "content": " satisfies", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [228, 483, 438, 495], "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "score": 0.88, "index": 84}, {"type": "text", "coordinates": [439, 481, 467, 497], "content": ", and ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [468, 484, 536, 494], "content": "\\kappa=k{+}2r{-}2", "score": 0.91, "index": 86}, {"type": "text", "coordinates": [537, 481, 540, 497], "content": ".", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [69, 495, 114, 513], "content": "For any ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [114, 501, 120, 507], "content": "r", "score": 0.82, "index": 89}, {"type": "text", "coordinates": [121, 495, 265, 513], "content": ", there are the conjugations ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [266, 498, 307, 509], "content": "C_{0}=i d", "score": 0.92, "index": 91}, {"type": "text", "coordinates": [307, 495, 335, 513], "content": ". and ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [335, 498, 510, 510], "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "score": 0.91, "index": 93}, {"type": "text", "coordinates": [511, 495, 541, 513], "content": ". The", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [70, 511, 173, 527], "content": "charge-conjugation ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [174, 513, 183, 522], "content": "C", "score": 0.87, "index": 96}, {"type": "text", "coordinates": [184, 511, 223, 527], "content": " equals ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [223, 513, 237, 524], "content": "C_{1}", "score": 0.91, "index": 98}, {"type": "text", "coordinates": [238, 511, 284, 527], "content": " for odd ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [284, 516, 290, 522], "content": "r", "score": 0.87, "index": 100}, {"type": "text", "coordinates": [291, 511, 321, 527], "content": ", and ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [321, 513, 335, 524], "content": "C_{0}", "score": 0.92, "index": 102}, {"type": "text", "coordinates": [335, 511, 385, 527], "content": " for even ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [385, 516, 391, 522], "content": "r", "score": 0.88, "index": 104}, {"type": "text", "coordinates": [391, 511, 434, 527], "content": ". When ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [435, 513, 464, 522], "content": "r=4", "score": 0.9, "index": 106}, {"type": "text", "coordinates": [464, 511, 541, 527], "content": " there are four", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [69, 526, 252, 542], "content": "additional conjugations; these six ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [253, 529, 265, 540], "content": "C_{i}", "score": 0.91, "index": 109}, {"type": "text", "coordinates": [265, 526, 473, 542], "content": " correspond to all permutations of the ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [473, 524, 496, 541], "content": "{D}_{4}^{(1)}", "score": 0.94, "index": 111}, {"type": "text", "coordinates": [497, 526, 541, 542], "content": "Dynkin", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [71, 542, 104, 557], "content": "labels ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [105, 543, 153, 554], "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "score": 0.92, "index": 114}, {"type": "text", "coordinates": [153, 542, 156, 557], "content": ".", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [93, 554, 335, 572], "content": "There are three non-trivial simple-currents, ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [335, 557, 347, 568], "content": "J_{v}", "score": 0.84, "index": 117}, {"type": "text", "coordinates": [348, 554, 356, 572], "content": ", ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [357, 558, 369, 568], "content": "J_{s}", "score": 0.81, "index": 119}, {"type": "text", "coordinates": [369, 554, 399, 572], "content": " and ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [399, 558, 423, 568], "content": "J_{v}J_{s}", "score": 0.91, "index": 121}, {"type": "text", "coordinates": [424, 554, 541, 572], "content": ". Explicitly, we have", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [71, 571, 262, 584], "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "score": 0.91, "index": 123}, {"type": "text", "coordinates": [263, 570, 293, 586], "content": " with ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [293, 571, 414, 584], "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "score": 0.9, "index": 125}, {"type": "text", "coordinates": [414, 570, 442, 586], "content": ", and", "score": 1.0, "index": 126}, {"type": "interline_equation", "coordinates": [174, 596, 430, 627], "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "score": 0.87, "index": 127}, {"type": "text", "coordinates": [69, 641, 97, 660], "content": "with ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [97, 639, 320, 658], "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "score": 0.91, "index": 129}, {"type": "text", "coordinates": [321, 641, 444, 660], "content": ". From this we compute", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [444, 642, 536, 656], "content": "Q_{s}(J_{s}0)=-r k/4", "score": 0.91, "index": 131}, {"type": "text", "coordinates": [536, 641, 541, 660], "content": ".", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [94, 657, 268, 672], "content": "The fusion products we need are", "score": 1.0, "index": 133}]	[]	[{"type": "block", "coordinates": [267, 129, 344, 146], "content": "", "caption": ""}, {"type": "block", "coordinates": [201, 234, 406, 251], "content": "", "caption": ""}, {"type": "block", "coordinates": [209, 289, 401, 329], "content": "", "caption": ""}, {"type": "block", "coordinates": [174, 596, 430, 627], "content": "", "caption": ""}, {"type": "inline", "coordinates": [303, 75, 311, 84], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [482, 75, 495, 86], "content": "C_{1}", "caption": ""}, {"type": "inline", "coordinates": [149, 90, 164, 101], "content": "A_{1}", "caption": ""}, {"type": "inline", "coordinates": [247, 87, 304, 101], "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "caption": ""}, {"type": "inline", "coordinates": [85, 104, 107, 117], "content": "C_{k,r}", "caption": ""}, {"type": "inline", "coordinates": [157, 103, 286, 117], "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "caption": ""}, {"type": "inline", "coordinates": [234, 158, 257, 174], "content": "C_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [127, 176, 225, 190], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "caption": ""}, {"type": "inline", "coordinates": [312, 176, 341, 188], "content": "k=r", "caption": ""}, {"type": "inline", "coordinates": [71, 192, 128, 208], "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "caption": ""}, {"type": "inline", "coordinates": [118, 267, 144, 276], "content": "i<r", "caption": ""}, {"type": "inline", "coordinates": [171, 264, 200, 277], "content": "k\\geq2", "caption": ""}, {"type": "inline", "coordinates": [105, 342, 203, 356], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "caption": ""}, {"type": "inline", "coordinates": [325, 362, 348, 378], "content": "C_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [378, 365, 387, 377], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [424, 365, 455, 379], "content": "k\\neq r", "caption": ""}, {"type": "inline", "coordinates": [516, 366, 524, 377], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [71, 385, 78, 391], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [146, 379, 171, 393], "content": "\\pi[1]^{i}", "caption": ""}, {"type": "inline", "coordinates": [193, 379, 243, 394], "content": "i\\in\\{0,1\\}", "caption": ""}, {"type": "inline", "coordinates": [286, 380, 317, 393], "content": "k\\neq r", "caption": ""}, {"type": "inline", "coordinates": [365, 380, 373, 391], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [399, 382, 407, 391], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [255, 394, 285, 406], "content": "k=r", "caption": ""}, {"type": "inline", "coordinates": [337, 392, 384, 409], "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "caption": ""}, {"type": "inline", "coordinates": [391, 393, 399, 406], "content": "\\mathit{\\Pi}_{k}", "caption": ""}, {"type": "inline", "coordinates": [450, 393, 475, 408], "content": "\\pi[1]^{i}", "caption": ""}, {"type": "inline", "coordinates": [483, 394, 491, 406], "content": "\\mathit{k}", "caption": ""}, {"type": "inline", "coordinates": [71, 409, 131, 422], "content": "i,j\\in\\{0,1\\}", "caption": ""}, {"type": "inline", "coordinates": [130, 431, 159, 441], "content": "r=k", "caption": ""}, {"type": "inline", "coordinates": [205, 429, 303, 444], "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "caption": ""}, {"type": "inline", "coordinates": [161, 458, 219, 474], "content": "D_{r}^{(1)},\\,r\\geq4", "caption": ""}, {"type": "inline", "coordinates": [144, 483, 151, 493], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [166, 484, 182, 496], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [228, 483, 438, 495], "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [468, 484, 536, 494], "content": "\\kappa=k{+}2r{-}2", "caption": ""}, {"type": "inline", "coordinates": [114, 501, 120, 507], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [266, 498, 307, 509], "content": "C_{0}=i d", "caption": ""}, {"type": "inline", "coordinates": [335, 498, 510, 510], "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "caption": ""}, {"type": "inline", "coordinates": [174, 513, 183, 522], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [223, 513, 237, 524], "content": "C_{1}", "caption": ""}, {"type": "inline", "coordinates": [284, 516, 290, 522], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [321, 513, 335, 524], "content": "C_{0}", "caption": ""}, {"type": "inline", "coordinates": [385, 516, 391, 522], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [435, 513, 464, 522], "content": "r=4", "caption": ""}, {"type": "inline", "coordinates": [253, 529, 265, 540], "content": "C_{i}", "caption": ""}, {"type": "inline", "coordinates": [473, 524, 496, 541], "content": "{D}_{4}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [105, 543, 153, 554], "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [335, 557, 347, 568], "content": "J_{v}", "caption": ""}, {"type": "inline", "coordinates": [357, 558, 369, 568], "content": "J_{s}", "caption": ""}, {"type": "inline", "coordinates": [399, 558, 423, 568], "content": "J_{v}J_{s}", "caption": ""}, {"type": "inline", "coordinates": [71, 571, 262, 584], "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "caption": ""}, {"type": "inline", "coordinates": [293, 571, 414, 584], "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "caption": ""}, {"type": "inline", "coordinates": [97, 639, 320, 658], "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [444, 642, 536, 656], "content": "Q_{s}(J_{s}0)=-r k/4", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 10}, {"type": "equation", "text": "$$\n\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "This rank-level duality for $C_{r}^{(1)}$ is especially interesting, as it defines a fusion ring isomorphism $\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$ (see \u00a75). When $k=r$ , we get a nontrivial fusion-symmetry: $\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda$ . ", "page_idx": 10}, {"type": "text", "text": "The only fusion product we need is ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "valid for $i<r$ and $k\\geq2$ . The following character formula (2.1b) will also be used: ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$ as before. ", "page_idx": 10}, {"type": "text", "text": "Theorem 3.C. The fusion-symmetries for $C_{r}^{(1)}$ level $k$ , when $k\\neq r$ and either $k$ or $r$ is even, are $\\pi[1]^{i}$ for $i\\in\\{0,1\\}$ . When $k\\neq r$ but both $k$ and $r$ are odd, then there is no nontrivial fusion-symmetry. When $k=r$ , they are $\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}$ $\\mathit{\\Pi}_{k}$ even) or $\\pi[1]^{i}$ ( $\\mathit{k}$ odd), for $i,j\\in\\{0,1\\}$ . ", "page_idx": 10}, {"type": "text", "text": "When $r=k$ is even, $A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$ . ", "page_idx": 10}, {"type": "text", "text": "3.4. The algebra $D_{r}^{(1)},\\,r\\geq4$ ", "page_idx": 10}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}$ , and $\\kappa=k{+}2r{-}2$ . For any $r$ , there are the conjugations $C_{0}=i d$ . and $C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})$ . The charge-conjugation $C$ equals $C_{1}$ for odd $r$ , and $C_{0}$ for even $r$ . When $r=4$ there are four additional conjugations; these six $C_{i}$ correspond to all permutations of the ${D}_{4}^{(1)}$ Dynkin labels $\\lambda_{1},\\lambda_{3},\\lambda_{4}$ . ", "page_idx": 10}, {"type": "text", "text": "There are three non-trivial simple-currents, $J_{v}$ , $J_{s}$ and $J_{v}J_{s}$ . Explicitly, we have $J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)$ with $Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2$ , and ", "page_idx": 10}, {"type": "equation", "text": "$$\nJ_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "with $\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}$ . From this we compute $Q_{s}(J_{s}0)=-r k/4$ . The fusion products we need are ", "page_idx": 10}]	[{"category_id": 1, "poly": [197, 1333, 1505, 1333, 1505, 1539, 197, 1539], "score": 0.976}, {"category_id": 1, "poly": [196, 1004, 1507, 1004, 1507, 1172, 196, 1172], "score": 0.974}, {"category_id": 1, "poly": [195, 196, 1506, 196, 1506, 321, 195, 321], "score": 0.971}, {"category_id": 1, "poly": [195, 439, 1505, 439, 1505, 571, 195, 571], "score": 0.966}, {"category_id": 1, "poly": [197, 1540, 1503, 1540, 1503, 1623, 197, 1623], "score": 0.952}, {"category_id": 8, "poly": [578, 804, 1118, 804, 1118, 911, 578, 911], "score": 0.948}, {"category_id": 8, "poly": [485, 1650, 1198, 1650, 1198, 1743, 485, 1743], "score": 0.946}, {"category_id": 1, "poly": [195, 943, 717, 943, 717, 987, 195, 987], "score": 0.943}, {"category_id": 1, "poly": [193, 1775, 1503, 1775, 1503, 1859, 193, 1859], "score": 0.941}, {"category_id": 1, "poly": [198, 728, 1404, 728, 1404, 771, 198, 771], "score": 0.936}, {"category_id": 8, "poly": [564, 647, 1135, 647, 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115], "lines": [{"bbox": [72, 74, 541, 88], "spans": [{"bbox": [72, 74, 303, 88], "score": 1.0, "content": "whose diagram is the transpose of that for ", "type": "text"}, {"bbox": [303, 75, 311, 84], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [311, 74, 481, 88], "score": 1.0, "content": ". (For this purpose the algebra ", "type": "text"}, {"bbox": [482, 75, 495, 86], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [496, 74, 541, 88], "score": 1.0, "content": " may be", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 103], "spans": [{"bbox": [70, 87, 149, 103], "score": 1.0, "content": "identified with ", "type": "text"}, {"bbox": [149, 90, 164, 101], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [164, 87, 246, 103], "score": 1.0, "content": ".) For example,", "type": "text"}, {"bbox": [247, 87, 304, 101], "score": 0.94, "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [304, 87, 541, 103], "score": 1.0, "content": ", where we use tilde\u2019s to denote the quantities", "type": "text"}], "index": 1}, {"bbox": [70, 102, 394, 117], "spans": [{"bbox": [70, 102, 84, 117], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 104, 107, 117], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [107, 102, 157, 117], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [157, 103, 286, 117], "score": 0.94, "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 129}, {"bbox": [286, 102, 394, 117], "score": 1.0, "content": " is a bijection. Then", "type": "text"}], "index": 2}], "index": 1}, {"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 158, 541, 205], "lines": [{"bbox": [93, 158, 540, 178], "spans": [{"bbox": [93, 159, 234, 178], "score": 1.0, "content": "This rank-level duality for ", "type": "text"}, {"bbox": [234, 158, 257, 174], "score": 0.91, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [258, 159, 540, 178], "score": 1.0, "content": "is especially interesting, as it defines a fusion ring iso-", "type": "text"}], "index": 4}, {"bbox": [71, 176, 540, 191], "spans": [{"bbox": [71, 176, 127, 191], "score": 1.0, "content": "morphism ", "type": "text"}, {"bbox": [127, 176, 225, 190], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [226, 176, 311, 191], "score": 1.0, "content": " (see \u00a75). When ", "type": "text"}, {"bbox": [312, 176, 341, 188], "score": 0.89, "content": "k=r", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [342, 176, 540, 191], "score": 1.0, "content": ", we get a nontrivial fusion-symmetry:", "type": "text"}], "index": 5}, {"bbox": [71, 189, 134, 210], "spans": [{"bbox": [71, 192, 128, 208], "score": 0.92, "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "type": "inline_equation", "height": 16, "width": 57}, {"bbox": [129, 189, 134, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [95, 207, 282, 221], "lines": [{"bbox": [95, 209, 281, 223], "spans": [{"bbox": [95, 209, 281, 223], "score": 1.0, "content": "The only fusion product we need is", "type": "text"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [71, 262, 505, 277], "lines": [{"bbox": [70, 263, 506, 281], "spans": [{"bbox": [70, 263, 118, 281], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 267, 144, 276], "score": 0.9, "content": "i<r", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [145, 263, 171, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [171, 264, 200, 277], "score": 0.9, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [201, 263, 506, 281], "score": 1.0, "content": ". The following character formula (2.1b) will also be used:", "type": "text"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 339, 258, 355], "lines": [{"bbox": [71, 341, 257, 357], "spans": [{"bbox": [71, 341, 105, 357], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 342, 203, 356], "score": 0.92, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [203, 341, 257, 357], "score": 1.0, "content": " as before.", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 361, 542, 421], "lines": [{"bbox": [93, 362, 542, 381], "spans": [{"bbox": [93, 362, 324, 381], "score": 1.0, "content": "Theorem 3.C. The fusion-symmetries for ", "type": "text"}, {"bbox": [325, 362, 348, 378], "score": 0.9, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [348, 362, 378, 381], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [378, 365, 387, 377], "score": 0.75, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [387, 362, 424, 381], "score": 1.0, "content": ", when ", "type": "text"}, {"bbox": [424, 365, 455, 379], "score": 0.9, "content": "k\\neq r", "type": "inline_equation", "height": 14, "width": 31}, {"bbox": [456, 362, 516, 381], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [516, 366, 524, 377], "score": 0.69, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [525, 362, 542, 381], "score": 1.0, "content": " or", "type": "text"}], "index": 12}, {"bbox": [71, 379, 541, 394], "spans": [{"bbox": [71, 385, 78, 391], "score": 0.66, "content": "r", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [78, 380, 146, 394], "score": 1.0, "content": " is even, are ", "type": "text"}, {"bbox": [146, 379, 171, 393], "score": 0.9, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [171, 380, 193, 394], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [193, 379, 243, 394], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [243, 380, 286, 394], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [286, 380, 317, 393], "score": 0.91, "content": "k\\neq r", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [317, 380, 365, 394], "score": 1.0, "content": " but both ", "type": "text"}, {"bbox": [365, 380, 373, 391], "score": 0.77, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [374, 380, 399, 394], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 382, 407, 391], "score": 0.69, "content": "r", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [407, 380, 541, 394], "score": 1.0, "content": " are odd, then there is no", "type": "text"}], "index": 13}, {"bbox": [70, 392, 542, 410], "spans": [{"bbox": [70, 394, 255, 410], "score": 1.0, "content": "nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [255, 394, 285, 406], "score": 0.88, "content": "k=r", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [285, 394, 337, 410], "score": 1.0, "content": ", they are ", "type": "text"}, {"bbox": [337, 392, 384, 409], "score": 0.91, "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "type": "inline_equation", "height": 17, "width": 47}, {"bbox": [384, 394, 390, 410], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [391, 393, 399, 406], "score": 0.6, "content": "\\mathit{\\Pi}_{k}", "type": "inline_equation", "height": 13, "width": 8}, {"bbox": [400, 394, 449, 410], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [450, 393, 475, 408], "score": 0.79, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [475, 394, 482, 410], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [483, 394, 491, 406], "score": 0.43, "content": "\\mathit{k}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [491, 394, 542, 410], "score": 1.0, "content": " odd), for", "type": "text"}], "index": 14}, {"bbox": [71, 407, 136, 423], "spans": [{"bbox": [71, 409, 131, 422], "score": 0.91, "content": "i,j\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 407, 136, 423], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [94, 427, 306, 443], "lines": [{"bbox": [95, 429, 305, 444], "spans": [{"bbox": [95, 429, 129, 444], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 431, 159, 441], "score": 0.88, "content": "r=k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [159, 429, 204, 444], "score": 1.0, "content": " is even, ", "type": "text"}, {"bbox": [205, 429, 303, 444], "score": 0.91, "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [303, 429, 305, 444], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [71, 456, 221, 474], "lines": [{"bbox": [70, 456, 219, 476], "spans": [{"bbox": [70, 456, 160, 476], "score": 1.0, "content": "3.4. The algebra ", "type": "text"}, {"bbox": [161, 458, 219, 474], "score": 0.3, "content": "D_{r}^{(1)},\\,r\\geq4", "type": "inline_equation", "height": 16, "width": 58}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 479, 541, 554], "lines": [{"bbox": [94, 481, 540, 497], "spans": [{"bbox": [94, 481, 144, 497], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 483, 151, 493], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [151, 481, 166, 497], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [166, 484, 182, 496], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [182, 481, 227, 497], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [228, 483, 438, 495], "score": 0.88, "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 210}, {"bbox": [439, 481, 467, 497], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [468, 484, 536, 494], "score": 0.91, "content": "\\kappa=k{+}2r{-}2", "type": "inline_equation", "height": 10, "width": 68}, {"bbox": [537, 481, 540, 497], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [69, 495, 541, 513], "spans": [{"bbox": [69, 495, 114, 513], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [114, 501, 120, 507], "score": 0.82, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [121, 495, 265, 513], "score": 1.0, "content": ", there are the conjugations ", "type": "text"}, {"bbox": [266, 498, 307, 509], "score": 0.92, "content": "C_{0}=i d", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [307, 495, 335, 513], "score": 1.0, "content": ". and ", "type": "text"}, {"bbox": [335, 498, 510, 510], "score": 0.91, "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "type": "inline_equation", "height": 12, "width": 175}, {"bbox": [511, 495, 541, 513], "score": 1.0, "content": ". The", "type": "text"}], "index": 19}, {"bbox": [70, 511, 541, 527], "spans": [{"bbox": [70, 511, 173, 527], "score": 1.0, "content": "charge-conjugation ", "type": "text"}, {"bbox": [174, 513, 183, 522], "score": 0.87, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 511, 223, 527], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [223, 513, 237, 524], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [238, 511, 284, 527], "score": 1.0, "content": " for odd ", "type": "text"}, {"bbox": [284, 516, 290, 522], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [291, 511, 321, 527], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [321, 513, 335, 524], "score": 0.92, "content": "C_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [335, 511, 385, 527], "score": 1.0, "content": " for even ", "type": "text"}, {"bbox": [385, 516, 391, 522], "score": 0.88, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [391, 511, 434, 527], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [435, 513, 464, 522], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [464, 511, 541, 527], "score": 1.0, "content": " there are four", "type": "text"}], "index": 20}, {"bbox": [69, 524, 541, 542], "spans": [{"bbox": [69, 526, 252, 542], "score": 1.0, "content": "additional conjugations; these six ", "type": "text"}, {"bbox": [253, 529, 265, 540], "score": 0.91, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [265, 526, 473, 542], "score": 1.0, "content": " correspond to all permutations of the ", "type": "text"}, {"bbox": [473, 524, 496, 541], "score": 0.94, "content": "{D}_{4}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [497, 526, 541, 542], "score": 1.0, "content": "Dynkin", "type": "text"}], "index": 21}, {"bbox": [71, 542, 156, 557], "spans": [{"bbox": [71, 542, 104, 557], "score": 1.0, "content": "labels ", "type": "text"}, {"bbox": [105, 543, 153, 554], "score": 0.92, "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [153, 542, 156, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 20}, {"type": "text", "bbox": [70, 554, 541, 584], "lines": [{"bbox": [93, 554, 541, 572], "spans": [{"bbox": [93, 554, 335, 572], "score": 1.0, "content": "There are three non-trivial simple-currents, ", "type": "text"}, {"bbox": [335, 557, 347, 568], "score": 0.84, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [348, 554, 356, 572], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [357, 558, 369, 568], "score": 0.81, "content": "J_{s}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [369, 554, 399, 572], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 558, 423, 568], "score": 0.91, "content": "J_{v}J_{s}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [424, 554, 541, 572], "score": 1.0, "content": ". Explicitly, we have", "type": "text"}], "index": 23}, {"bbox": [71, 570, 442, 586], "spans": [{"bbox": [71, 571, 262, 584], "score": 0.91, "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 13, "width": 191}, {"bbox": [263, 570, 293, 586], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 571, 414, 584], "score": 0.9, "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "type": "inline_equation", "height": 13, "width": 121}, {"bbox": [414, 570, 442, 586], "score": 1.0, "content": ", and", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [69, 639, 541, 669], "lines": [{"bbox": [69, 639, 541, 660], "spans": [{"bbox": [69, 641, 97, 660], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [97, 639, 320, 658], "score": 0.91, "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "type": "inline_equation", "height": 19, "width": 223}, {"bbox": [321, 641, 444, 660], "score": 1.0, "content": ". From this we compute", "type": "text"}, {"bbox": [444, 642, 536, 656], "score": 0.91, "content": "Q_{s}(J_{s}0)=-r k/4", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [536, 641, 541, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [94, 657, 268, 672], "spans": [{"bbox": [94, 657, 268, 672], "score": 1.0, "content": "The fusion products we need are", "type": "text"}], "index": 27}], "index": 26.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 542, 115], "lines": [], "index": 1, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 74, 541, 117], "lines_deleted": true}, {"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 158, 541, 205], "lines": [{"bbox": [93, 158, 540, 178], "spans": [{"bbox": [93, 159, 234, 178], "score": 1.0, "content": "This rank-level duality for ", "type": "text"}, {"bbox": [234, 158, 257, 174], "score": 0.91, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [258, 159, 540, 178], "score": 1.0, "content": "is especially interesting, as it defines a fusion ring iso-", "type": "text"}], "index": 4}, {"bbox": [71, 176, 540, 191], "spans": [{"bbox": [71, 176, 127, 191], "score": 1.0, "content": "morphism ", "type": "text"}, {"bbox": [127, 176, 225, 190], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [226, 176, 311, 191], "score": 1.0, "content": " (see \u00a75). When ", "type": "text"}, {"bbox": [312, 176, 341, 188], "score": 0.89, "content": "k=r", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [342, 176, 540, 191], "score": 1.0, "content": ", we get a nontrivial fusion-symmetry:", "type": "text"}], "index": 5}, {"bbox": [71, 189, 134, 210], "spans": [{"bbox": [71, 192, 128, 208], "score": 0.92, "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "type": "inline_equation", "height": 16, "width": 57}, {"bbox": [129, 189, 134, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [71, 158, 540, 210]}, {"type": "text", "bbox": [95, 207, 282, 221], "lines": [{"bbox": [95, 209, 281, 223], "spans": [{"bbox": [95, 209, 281, 223], "score": 1.0, "content": "The only fusion product we need is", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [95, 209, 281, 223]}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 262, 505, 277], "lines": [{"bbox": [70, 263, 506, 281], "spans": [{"bbox": [70, 263, 118, 281], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 267, 144, 276], "score": 0.9, "content": "i<r", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [145, 263, 171, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [171, 264, 200, 277], "score": 0.9, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [201, 263, 506, 281], "score": 1.0, "content": ". The following character formula (2.1b) will also be used:", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 263, 506, 281]}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 339, 258, 355], "lines": [{"bbox": [71, 341, 257, 357], "spans": [{"bbox": [71, 341, 105, 357], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 342, 203, 356], "score": 0.92, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [203, 341, 257, 357], "score": 1.0, "content": " as before.", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [71, 341, 257, 357]}, {"type": "text", "bbox": [70, 361, 542, 421], "lines": [{"bbox": [93, 362, 542, 381], "spans": [{"bbox": [93, 362, 324, 381], "score": 1.0, "content": "Theorem 3.C. The fusion-symmetries for ", "type": "text"}, {"bbox": [325, 362, 348, 378], "score": 0.9, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [348, 362, 378, 381], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [378, 365, 387, 377], "score": 0.75, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [387, 362, 424, 381], "score": 1.0, "content": ", when ", "type": "text"}, {"bbox": [424, 365, 455, 379], "score": 0.9, "content": "k\\neq r", "type": "inline_equation", "height": 14, "width": 31}, {"bbox": [456, 362, 516, 381], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [516, 366, 524, 377], "score": 0.69, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [525, 362, 542, 381], "score": 1.0, "content": " or", "type": "text"}], "index": 12}, {"bbox": [71, 379, 541, 394], "spans": [{"bbox": [71, 385, 78, 391], "score": 0.66, "content": "r", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [78, 380, 146, 394], "score": 1.0, "content": " is even, are ", "type": "text"}, {"bbox": [146, 379, 171, 393], "score": 0.9, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [171, 380, 193, 394], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [193, 379, 243, 394], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [243, 380, 286, 394], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [286, 380, 317, 393], "score": 0.91, "content": "k\\neq r", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [317, 380, 365, 394], "score": 1.0, "content": " but both ", "type": "text"}, {"bbox": [365, 380, 373, 391], "score": 0.77, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [374, 380, 399, 394], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 382, 407, 391], "score": 0.69, "content": "r", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [407, 380, 541, 394], "score": 1.0, "content": " are odd, then there is no", "type": "text"}], "index": 13}, {"bbox": [70, 392, 542, 410], "spans": [{"bbox": [70, 394, 255, 410], "score": 1.0, "content": "nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [255, 394, 285, 406], "score": 0.88, "content": "k=r", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [285, 394, 337, 410], "score": 1.0, "content": ", they are ", "type": "text"}, {"bbox": [337, 392, 384, 409], "score": 0.91, "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "type": "inline_equation", "height": 17, "width": 47}, {"bbox": [384, 394, 390, 410], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [391, 393, 399, 406], "score": 0.6, "content": "\\mathit{\\Pi}_{k}", "type": "inline_equation", "height": 13, "width": 8}, {"bbox": [400, 394, 449, 410], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [450, 393, 475, 408], "score": 0.79, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [475, 394, 482, 410], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [483, 394, 491, 406], "score": 0.43, "content": "\\mathit{k}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [491, 394, 542, 410], "score": 1.0, "content": " odd), for", "type": "text"}], "index": 14}, {"bbox": [71, 407, 136, 423], "spans": [{"bbox": [71, 409, 131, 422], "score": 0.91, "content": "i,j\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 407, 136, 423], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 362, 542, 423]}, {"type": "text", "bbox": [94, 427, 306, 443], "lines": [{"bbox": [95, 429, 305, 444], "spans": [{"bbox": [95, 429, 129, 444], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 431, 159, 441], "score": 0.88, "content": "r=k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [159, 429, 204, 444], "score": 1.0, "content": " is even, ", "type": "text"}, {"bbox": [205, 429, 303, 444], "score": 0.91, "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [303, 429, 305, 444], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [95, 429, 305, 444]}, {"type": "text", "bbox": [71, 456, 221, 474], "lines": [{"bbox": [70, 456, 219, 476], "spans": [{"bbox": [70, 456, 160, 476], "score": 1.0, "content": "3.4. The algebra ", "type": "text"}, {"bbox": [161, 458, 219, 474], "score": 0.3, "content": "D_{r}^{(1)},\\,r\\geq4", "type": "inline_equation", "height": 16, "width": 58}], "index": 17}], "index": 17, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 456, 219, 476]}, {"type": "text", "bbox": [70, 479, 541, 554], "lines": [{"bbox": [94, 481, 540, 497], "spans": [{"bbox": [94, 481, 144, 497], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 483, 151, 493], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [151, 481, 166, 497], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [166, 484, 182, 496], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [182, 481, 227, 497], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [228, 483, 438, 495], "score": 0.88, "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 210}, {"bbox": [439, 481, 467, 497], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [468, 484, 536, 494], "score": 0.91, "content": "\\kappa=k{+}2r{-}2", "type": "inline_equation", "height": 10, "width": 68}, {"bbox": [537, 481, 540, 497], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [69, 495, 541, 513], "spans": [{"bbox": [69, 495, 114, 513], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [114, 501, 120, 507], "score": 0.82, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [121, 495, 265, 513], "score": 1.0, "content": ", there are the conjugations ", "type": "text"}, {"bbox": [266, 498, 307, 509], "score": 0.92, "content": "C_{0}=i d", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [307, 495, 335, 513], "score": 1.0, "content": ". and ", "type": "text"}, {"bbox": [335, 498, 510, 510], "score": 0.91, "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "type": "inline_equation", "height": 12, "width": 175}, {"bbox": [511, 495, 541, 513], "score": 1.0, "content": ". The", "type": "text"}], "index": 19}, {"bbox": [70, 511, 541, 527], "spans": [{"bbox": [70, 511, 173, 527], "score": 1.0, "content": "charge-conjugation ", "type": "text"}, {"bbox": [174, 513, 183, 522], "score": 0.87, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 511, 223, 527], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [223, 513, 237, 524], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [238, 511, 284, 527], "score": 1.0, "content": " for odd ", "type": "text"}, {"bbox": [284, 516, 290, 522], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [291, 511, 321, 527], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [321, 513, 335, 524], "score": 0.92, "content": "C_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [335, 511, 385, 527], "score": 1.0, "content": " for even ", "type": "text"}, {"bbox": [385, 516, 391, 522], "score": 0.88, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [391, 511, 434, 527], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [435, 513, 464, 522], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [464, 511, 541, 527], "score": 1.0, "content": " there are four", "type": "text"}], "index": 20}, {"bbox": [69, 524, 541, 542], "spans": [{"bbox": [69, 526, 252, 542], "score": 1.0, "content": "additional conjugations; these six ", "type": "text"}, {"bbox": [253, 529, 265, 540], "score": 0.91, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [265, 526, 473, 542], "score": 1.0, "content": " correspond to all permutations of the ", "type": "text"}, {"bbox": [473, 524, 496, 541], "score": 0.94, "content": "{D}_{4}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [497, 526, 541, 542], "score": 1.0, "content": "Dynkin", "type": "text"}], "index": 21}, {"bbox": [71, 542, 156, 557], "spans": [{"bbox": [71, 542, 104, 557], "score": 1.0, "content": "labels ", "type": "text"}, {"bbox": [105, 543, 153, 554], "score": 0.92, "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [153, 542, 156, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 20, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [69, 481, 541, 557]}, {"type": "text", "bbox": [70, 554, 541, 584], "lines": [{"bbox": [93, 554, 541, 572], "spans": [{"bbox": [93, 554, 335, 572], "score": 1.0, "content": "There are three non-trivial simple-currents, ", "type": "text"}, {"bbox": [335, 557, 347, 568], "score": 0.84, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [348, 554, 356, 572], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [357, 558, 369, 568], "score": 0.81, "content": "J_{s}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [369, 554, 399, 572], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 558, 423, 568], "score": 0.91, "content": "J_{v}J_{s}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [424, 554, 541, 572], "score": 1.0, "content": ". Explicitly, we have", "type": "text"}], "index": 23}, {"bbox": [71, 570, 442, 586], "spans": [{"bbox": [71, 571, 262, 584], "score": 0.91, "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 13, "width": 191}, {"bbox": [263, 570, 293, 586], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 571, 414, 584], "score": 0.9, "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "type": "inline_equation", "height": 13, "width": 121}, {"bbox": [414, 570, 442, 586], "score": 1.0, "content": ", and", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [71, 554, 541, 586]}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 639, 541, 669], "lines": [{"bbox": [69, 639, 541, 660], "spans": [{"bbox": [69, 641, 97, 660], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [97, 639, 320, 658], "score": 0.91, "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "type": "inline_equation", "height": 19, "width": 223}, {"bbox": [321, 641, 444, 660], "score": 1.0, "content": ". From this we compute", "type": "text"}, {"bbox": [444, 642, 536, 656], "score": 0.91, "content": "Q_{s}(J_{s}0)=-r k/4", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [536, 641, 541, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [94, 657, 268, 672], "spans": [{"bbox": [94, 657, 268, 672], "score": 1.0, "content": "The fusion products we need are", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [69, 639, 541, 672]}]}
0002044v1	12	$${D}_{4}^{(1)}$$ at $$k=2$$ . Finally, when both $$k=2$$ and $$r>4$$ , any fusion-symmetry $$\pi$$ can be written as $$\pi=C_{1}^{a}\,\pi_{v}^{b}\,\pi\{m\}$$ for $$a,b\in\{0,1\}$$ and any $$m\in\mathbb{Z}_{2r}^{\times}$$ , $$1\leq m<r$$ . $$\pi_{v}$$ here refers to the simple-current automorphism $$\pi[2]$$ or 10 00 ], for r odd/even. When $$k\,=\,1$$ , $$A(D_{e v e n,1})\cong{\mathfrak{S}}_{3}$$ , corresponding to any permutation of $$\Lambda_{1},\Lambda_{r-1},\Lambda_{r}$$ , and $$A(D_{o d d,1})\:=\:\langle{C_{1}}\rangle\:\cong\:\mathbb{Z}_{2}$$ . When $$r\,>\,4$$ , $${\cal A}(D_{r,2})\,\cong\,(\mathbb{Z}_{2r}^{\times}/\{\pm1\})\,\times\,\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2}$$ or $$\mathbb{Z}_{r}^{\times}\times\mathbb{Z}_{2}$$ for $$r$$ even/odd. $$A(D_{4,2})$$ has 24 elements, and any element can be written uniquely as $$C_{i}\,\pi\,\left[\begin{array}{l l}{a}&{0}\\ {0}&{d}\end{array}\right]$$ # 3.5. The algebra $$E_{6}^{(1)}$$ A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!3\lambda_{3}\!+\!2\lambda_{4}\!+\!\lambda_{5}\!+\!2\lambda_{6}$$ and $$\kappa=k\!+\!12$$ . The charge-conjugation acts as $$C\lambda=(\lambda_{0},\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{6})$$ . The order 3 simple-current $$J$$ is given by $$J\lambda=(\lambda_{5},\lambda_{0},\lambda_{6},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{4})$$ with $$Q(\lambda)=(-\lambda_{1}+\lambda_{2}-\lambda_{4}+\lambda_{5})/3$$ . The fusion products we need can be derived from [29] using (2.4): where the outer subscript on any summand denotes the smallest level where that sum- mand appears (it will also appear at all larger levels). So for example $$\Lambda_{1}\boxtimes\Lambda_{1}$$ equals $$\Lambda_{2}$$ + $$\Lambda_{5}$$ + $$(2\Lambda_{1})$$ for any $$k\geq2$$ , but equals $$\Lambda_{5}$$ at $$k\,=\,1$$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient). Theorem 3.E6. The fusion-symmetries of $$E_{6}^{(1)}$$ are $$C^{i}\,\pi[a]$$ , for any $$i\in\{0,1\}$$ and any $$a\in\{0,1,2\}$$ for which $$a k\not\equiv1$$ (mod 3). # 3.6. The algebra E7(1) A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k\,=\,\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+3\lambda_{4}+2\lambda_{5}+\lambda_{6}+2\lambda_{7}$$ , and $$\kappa\,=\,k\,+\,18$$ . The charge-conjugation is trivial, but there is a simple-current $$J$$ given by $$J\lambda=(\lambda_{6},\lambda_{5},...\,,\lambda_{1},\lambda_{0},\lambda_{7})$$ . It has $$Q(\lambda)=(\lambda_{4}+\lambda_{6}+\lambda_{7})/2$$ . The only fusion products we need can be obtained from [29] and (2.4): $$\Lambda_{6}$$ × $$\Lambda_{6}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{5})_{2}$$ + $$(2\Lambda_{6})_{2}$$ $$\Lambda_{1}$$ × $$\Lambda_{6}=(\Lambda_{6})_{2}$$ + $$(\Lambda_{7})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ $$\Lambda_{5}$$ × $$\Lambda_{6}=(\Lambda_{4})_{3}$$ + $$(\Lambda_{6})_{2}$$ + $$(\Lambda_{7})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ + $$(\Lambda_{5}+\Lambda_{6})_{3}$$ $$\Lambda_{6}$$ × $$(2\Lambda_{6})=(\Lambda_{6})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ + $$(3\Lambda_{6})_{3}$$ + $$(\Lambda_{5}+\Lambda_{6})_{3}$$ $$\Lambda_{4}$$ × $$\Lambda_{6}=(\Lambda_{2})_{3}$$ + $$(\Lambda_{3})_{4}$$ + $$(\Lambda_{5})_{3}$$ + $$(\Lambda_{1}+\Lambda_{5})_{4}$$ + $$(\Lambda_{4}+\Lambda_{6})_{4}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ $$\Lambda_{6}$$ × $$\Lambda_{7}=(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{3}$$ + $$(\Lambda_{5})_{2}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ $$\Lambda_{6}$$ × $$(\Lambda_{5}+\Lambda_{6})=(\Lambda_{5})_{3}$$ + $$(2\Lambda_{5})_{4}$$ + $$(2\Lambda_{6})_{3}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ + $$(\Lambda_{1}+\Lambda_{5})_{4}$$ + $$(\Lambda_{4}+\Lambda_{6})_{4}$$ + $$(\Lambda_{1}+2\Lambda_{6})_{4}$$ + $$(\Lambda_{5}+2\Lambda_{6})_{4}$$	<p>$${D}_{4}^{(1)}$$ at $$k=2$$ . Finally, when both $$k=2$$ and $$r>4$$ , any fusion-symmetry $$\pi$$ can be written as $$\pi=C_{1}^{a}\,\pi_{v}^{b}\,\pi\{m\}$$ for $$a,b\in\{0,1\}$$ and any $$m\in\mathbb{Z}_{2r}^{\times}$$ , $$1\leq m<r$$ .</p> <p>$$\pi_{v}$$ here refers to the simple-current automorphism $$\pi[2]$$ or 10 00 ], for r odd/even. When $$k\,=\,1$$ , $$A(D_{e v e n,1})\cong{\mathfrak{S}}_{3}$$ , corresponding to any permutation of $$\Lambda_{1},\Lambda_{r-1},\Lambda_{r}$$ , and $$A(D_{o d d,1})\:=\:\langle{C_{1}}\rangle\:\cong\:\mathbb{Z}_{2}$$ . When $$r\,>\,4$$ , $${\cal A}(D_{r,2})\,\cong\,(\mathbb{Z}_{2r}^{\times}/\{\pm1\})\,\times\,\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2}$$ or $$\mathbb{Z}_{r}^{\times}\times\mathbb{Z}_{2}$$ for $$r$$ even/odd. $$A(D_{4,2})$$ has 24 elements, and any element can be written uniquely as $$C_{i}\,\pi\,\left[\begin{array}{l l}{a}&{0}\\ {0}&{d}\end{array}\right]$$</p> <h1>3.5. The algebra $$E_{6}^{(1)}$$</h1> <p>A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!3\lambda_{3}\!+\!2\lambda_{4}\!+\!\lambda_{5}\!+\!2\lambda_{6}$$ and $$\kappa=k\!+\!12$$ . The charge-conjugation acts as $$C\lambda=(\lambda_{0},\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{6})$$ . The order 3 simple-current $$J$$ is given by $$J\lambda=(\lambda_{5},\lambda_{0},\lambda_{6},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{4})$$ with $$Q(\lambda)=(-\lambda_{1}+\lambda_{2}-\lambda_{4}+\lambda_{5})/3$$ .</p> <p>The fusion products we need can be derived from [29] using (2.4):</p> <p>where the outer subscript on any summand denotes the smallest level where that sum- mand appears (it will also appear at all larger levels). So for example $$\Lambda_{1}\boxtimes\Lambda_{1}$$ equals $$\Lambda_{2}$$ + $$\Lambda_{5}$$ + $$(2\Lambda_{1})$$ for any $$k\geq2$$ , but equals $$\Lambda_{5}$$ at $$k\,=\,1$$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient).</p> <p>Theorem 3.E6. The fusion-symmetries of $$E_{6}^{(1)}$$ are $$C^{i}\,\pi[a]$$ , for any $$i\in\{0,1\}$$ and any $$a\in\{0,1,2\}$$ for which $$a k\not\equiv1$$ (mod 3).</p> <h1>3.6. The algebra E7(1)</h1> <p>A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k\,=\,\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+3\lambda_{4}+2\lambda_{5}+\lambda_{6}+2\lambda_{7}$$ , and $$\kappa\,=\,k\,+\,18$$ . The charge-conjugation is trivial, but there is a simple-current $$J$$ given by $$J\lambda=(\lambda_{6},\lambda_{5},...\,,\lambda_{1},\lambda_{0},\lambda_{7})$$ . It has $$Q(\lambda)=(\lambda_{4}+\lambda_{6}+\lambda_{7})/2$$ .</p> <p>The only fusion products we need can be obtained from [29] and (2.4):</p> <p>$$\Lambda_{6}$$ × $$\Lambda_{6}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{5})_{2}$$ + $$(2\Lambda_{6})_{2}$$ $$\Lambda_{1}$$ × $$\Lambda_{6}=(\Lambda_{6})_{2}$$ + $$(\Lambda_{7})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ $$\Lambda_{5}$$ × $$\Lambda_{6}=(\Lambda_{4})_{3}$$ + $$(\Lambda_{6})_{2}$$ + $$(\Lambda_{7})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ + $$(\Lambda_{5}+\Lambda_{6})_{3}$$ $$\Lambda_{6}$$ × $$(2\Lambda_{6})=(\Lambda_{6})_{2}$$ + $$(\Lambda_{1}+\Lambda_{6})_{3}$$ + $$(3\Lambda_{6})_{3}$$ + $$(\Lambda_{5}+\Lambda_{6})_{3}$$ $$\Lambda_{4}$$ × $$\Lambda_{6}=(\Lambda_{2})_{3}$$ + $$(\Lambda_{3})_{4}$$ + $$(\Lambda_{5})_{3}$$ + $$(\Lambda_{1}+\Lambda_{5})_{4}$$ + $$(\Lambda_{4}+\Lambda_{6})_{4}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ $$\Lambda_{6}$$ × $$\Lambda_{7}=(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{3}$$ + $$(\Lambda_{5})_{2}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ $$\Lambda_{6}$$ × $$(\Lambda_{5}+\Lambda_{6})=(\Lambda_{5})_{3}$$ + $$(2\Lambda_{5})_{4}$$ + $$(2\Lambda_{6})_{3}$$ + $$(\Lambda_{6}+\Lambda_{7})_{3}$$ + $$(\Lambda_{1}+\Lambda_{5})_{4}$$ + $$(\Lambda_{4}+\Lambda_{6})_{4}$$ + $$(\Lambda_{1}+2\Lambda_{6})_{4}$$ + $$(\Lambda_{5}+2\Lambda_{6})_{4}$$</p>	[{"type": "text", "coordinates": [69, 69, 542, 102], "content": "$${D}_{4}^{(1)}$$ at $$k=2$$ . Finally, when both $$k=2$$ and $$r>4$$ , any fusion-symmetry $$\\pi$$ can be written\nas $$\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}$$ for $$a,b\\in\\{0,1\\}$$ and any $$m\\in\\mathbb{Z}_{2r}^{\\times}$$ , $$1\\leq m<r$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 106, 541, 204], "content": "$$\\pi_{v}$$ here refers to the simple-current automorphism $$\\pi[2]$$ or 10 00 ], for r odd/even.\nWhen $$k\\,=\\,1$$ , $$A(D_{e v e n,1})\\cong{\\mathfrak{S}}_{3}$$ , corresponding to any permutation of $$\\Lambda_{1},\\Lambda_{r-1},\\Lambda_{r}$$ , and\n$$A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}$$ . When $$r\\,>\\,4$$ , $${\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}$$ or $$\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}$$\nfor $$r$$ even/odd. $$A(D_{4,2})$$ has 24 elements, and any element can be written uniquely as\n$$C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]$$", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [70, 216, 183, 232], "content": "3.5. The algebra $$E_{6}^{(1)}$$", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [70, 237, 541, 280], "content": "A weight $$\\lambda$$ of $$P_{+}$$ satisfies $$k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}$$ and $$\\kappa=k\\!+\\!12$$ . The\ncharge-conjugation acts as $$C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})$$ . The order 3 simple-current $$J$$\nis given by $$J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})$$ with $$Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [96, 281, 443, 295], "content": "The fusion products we need can be derived from [29] using (2.4):", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [282, 354, 416, 370], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [70, 372, 541, 445], "content": "where the outer subscript on any summand denotes the smallest level where that sum-\nmand appears (it will also appear at all larger levels). So for example $$\\Lambda_{1}\\boxtimes\\Lambda_{1}$$ equals\n$$\\Lambda_{2}$$ + $$\\Lambda_{5}$$ + $$(2\\Lambda_{1})$$ for any $$k\\geq2$$ , but equals $$\\Lambda_{5}$$ at $$k\\,=\\,1$$ . A similar convention is used\nin (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will\nequal the numerical value of the fusion coefficient).", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [71, 450, 542, 482], "content": "Theorem 3.E6. The fusion-symmetries of $$E_{6}^{(1)}$$ are $$C^{i}\\,\\pi[a]$$ , for any $$i\\in\\{0,1\\}$$ and\nany $$a\\in\\{0,1,2\\}$$ for which $$a k\\not\\equiv1$$ (mod 3).", "block_type": "text", "index": 8}, {"type": "title", "coordinates": [71, 491, 183, 507], "content": "3.6. The algebra E7(1)", "block_type": "title", "index": 9}, {"type": "text", "coordinates": [70, 512, 542, 556], "content": "A weight $$\\lambda$$ in $$P_{+}$$ satisfies $$k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}$$ , and\n$$\\kappa\\,=\\,k\\,+\\,18$$ . The charge-conjugation is trivial, but there is a simple-current $$J$$ given by\n$$J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})$$ . It has $$Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [94, 556, 468, 570], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [82, 573, 530, 716], "content": "$$\\Lambda_{6}$$ \u00d7 $$\\Lambda_{6}=(0)_{1}$$ + $$(\\Lambda_{1})_{2}$$ + $$(\\Lambda_{5})_{2}$$ + $$(2\\Lambda_{6})_{2}$$\n$$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{6}=(\\Lambda_{6})_{2}$$ + $$(\\Lambda_{7})_{2}$$ + $$(\\Lambda_{1}+\\Lambda_{6})_{3}$$\n$$\\Lambda_{5}$$ \u00d7 $$\\Lambda_{6}=(\\Lambda_{4})_{3}$$ + $$(\\Lambda_{6})_{2}$$ + $$(\\Lambda_{7})_{2}$$ + $$(\\Lambda_{1}+\\Lambda_{6})_{3}$$ + $$(\\Lambda_{5}+\\Lambda_{6})_{3}$$\n$$\\Lambda_{6}$$ \u00d7 $$(2\\Lambda_{6})=(\\Lambda_{6})_{2}$$ + $$(\\Lambda_{1}+\\Lambda_{6})_{3}$$ + $$(3\\Lambda_{6})_{3}$$ + $$(\\Lambda_{5}+\\Lambda_{6})_{3}$$\n$$\\Lambda_{4}$$ \u00d7 $$\\Lambda_{6}=(\\Lambda_{2})_{3}$$ + $$(\\Lambda_{3})_{4}$$ + $$(\\Lambda_{5})_{3}$$ + $$(\\Lambda_{1}+\\Lambda_{5})_{4}$$ + $$(\\Lambda_{4}+\\Lambda_{6})_{4}$$ + $$(\\Lambda_{6}+\\Lambda_{7})_{3}$$\n$$\\Lambda_{6}$$ \u00d7 $$\\Lambda_{7}=(\\Lambda_{1})_{2}$$ + $$(\\Lambda_{2})_{3}$$ + $$(\\Lambda_{5})_{2}$$ + $$(\\Lambda_{6}+\\Lambda_{7})_{3}$$\n$$\\Lambda_{6}$$ \u00d7 $$(\\Lambda_{5}+\\Lambda_{6})=(\\Lambda_{5})_{3}$$ + $$(2\\Lambda_{5})_{4}$$ + $$(2\\Lambda_{6})_{3}$$ + $$(\\Lambda_{6}+\\Lambda_{7})_{3}$$ + $$(\\Lambda_{1}+\\Lambda_{5})_{4}$$\n+ $$(\\Lambda_{4}+\\Lambda_{6})_{4}$$ + $$(\\Lambda_{1}+2\\Lambda_{6})_{4}$$ + $$(\\Lambda_{5}+2\\Lambda_{6})_{4}$$", "block_type": "text", "index": 12}]	[{"type": "inline_equation", "coordinates": [71, 70, 95, 88], "content": "{D}_{4}^{(1)}", "score": 0.9, "index": 1}, {"type": "text", "coordinates": [95, 67, 112, 92], "content": "at ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [112, 74, 142, 86], "content": "k=2", "score": 0.86, "index": 3}, {"type": "text", "coordinates": [142, 67, 249, 92], "content": ". Finally, when both", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [250, 74, 280, 86], "content": "k=2", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [280, 67, 306, 92], "content": " and ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [306, 74, 335, 86], "content": "r>4", "score": 0.87, "index": 7}, {"type": "text", "coordinates": [335, 67, 454, 92], "content": ", any fusion-symmetry ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [454, 77, 463, 85], "content": "\\pi", "score": 0.63, "index": 9}, {"type": "text", "coordinates": [463, 67, 544, 92], "content": " can be written", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [70, 86, 86, 106], "content": "as ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [86, 87, 171, 102], "content": "\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}", "score": 0.89, "index": 12}, {"type": "text", "coordinates": [172, 86, 194, 106], "content": " for ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [194, 88, 255, 102], "content": "a,b\\in\\{0,1\\}", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [256, 86, 305, 106], "content": " and any ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [305, 87, 349, 102], "content": "m\\in\\mathbb{Z}_{2r}^{\\times}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [349, 86, 356, 106], "content": ", ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [356, 88, 411, 102], "content": "1\\leq m<r", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [412, 86, 418, 106], "content": ".", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [95, 116, 108, 127], "content": "\\pi_{v}", "score": 0.82, "index": 20}, {"type": "text", "coordinates": [108, 110, 367, 133], "content": " here refers to the simple-current automorphism ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [367, 113, 389, 128], "content": "\\pi[2]", "score": 0.26, "index": 22}, {"type": "text", "coordinates": [389, 110, 406, 133], "content": " or", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [420, 105, 545, 141], "content": " 10 00 ], for r odd/even.", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [70, 133, 106, 150], "content": "When ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [106, 134, 139, 146], "content": "k\\,=\\,1", "score": 0.86, "index": 26}, {"type": "text", "coordinates": [140, 133, 147, 150], "content": ", ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [147, 133, 239, 148], "content": "A(D_{e v e n,1})\\cong{\\mathfrak{S}}_{3}", "score": 0.9, "index": 28}, {"type": "text", "coordinates": [239, 133, 447, 150], "content": ", corresponding to any permutation of ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [447, 134, 512, 147], "content": "\\Lambda_{1},\\Lambda_{r-1},\\Lambda_{r}", "score": 0.93, "index": 30}, {"type": "text", "coordinates": [512, 133, 542, 150], "content": ", and", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [71, 148, 200, 163], "content": "A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [201, 148, 248, 166], "content": ". When ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [248, 149, 282, 161], "content": "r\\,>\\,4", "score": 0.85, "index": 34}, {"type": "text", "coordinates": [282, 148, 290, 166], "content": ", ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [290, 147, 473, 163], "content": "{\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}", "score": 0.91, "index": 36}, {"type": "text", "coordinates": [473, 148, 492, 166], "content": " or ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [493, 148, 540, 162], "content": "\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [70, 163, 90, 178], "content": "for ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [90, 166, 97, 174], "content": "r", "score": 0.65, "index": 40}, {"type": "text", "coordinates": [97, 163, 162, 178], "content": " even/odd. ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [162, 163, 204, 177], "content": "A(D_{4,2})", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [205, 163, 541, 178], "content": " has 24 elements, and any element can be written uniquely as", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [71, 176, 137, 208], "content": "C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]", "score": 0.89, "index": 44}, {"type": "text", "coordinates": [68, 217, 160, 237], "content": "3.5. The algebra ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [161, 217, 183, 234], "content": "E_{6}^{(1)}", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [95, 239, 144, 255], "content": "A weight ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [144, 240, 152, 250], "content": "\\lambda", "score": 0.8, "index": 48}, {"type": "text", "coordinates": [153, 239, 167, 255], "content": " of ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [167, 240, 183, 253], "content": "P_{+}", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [184, 239, 228, 255], "content": " satisfies", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [228, 240, 432, 252], "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [432, 239, 457, 255], "content": " and ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [457, 241, 510, 251], "content": "\\kappa=k\\!+\\!12", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [511, 239, 541, 255], "content": ". The", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [72, 253, 213, 269], "content": "charge-conjugation acts as ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [213, 253, 375, 267], "content": "C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})", "score": 0.89, "index": 57}, {"type": "text", "coordinates": [376, 253, 531, 269], "content": ". The order 3 simple-current ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [531, 256, 540, 265], "content": "J", "score": 0.87, "index": 59}, {"type": "text", "coordinates": [69, 268, 131, 284], "content": "is given by ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [131, 268, 290, 282], "content": "J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [290, 268, 319, 284], "content": " with", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [320, 267, 487, 282], "content": "Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3", "score": 0.91, "index": 63}, {"type": "text", "coordinates": [487, 268, 490, 284], "content": ".", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [95, 282, 440, 297], "content": "The fusion products we need can be derived from [29] using (2.4):", "score": 1.0, "index": 65}, {"type": "interline_equation", "coordinates": [282, 354, 416, 370], "content": "(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}", "score": 0.38, "index": 66}, {"type": "text", "coordinates": [70, 374, 540, 390], "content": "where the outer subscript on any summand denotes the smallest level where that sum-", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [70, 389, 455, 403], "content": "mand appears (it will also appear at all larger levels). So for example ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [456, 388, 502, 402], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{1}", "score": 0.36, "index": 69}, {"type": "text", "coordinates": [503, 389, 541, 403], "content": " equals", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [71, 404, 85, 416], "content": "\\Lambda_{2}", "score": 0.86, "index": 71}, {"type": "text", "coordinates": [86, 402, 102, 420], "content": " + ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [102, 403, 117, 416], "content": "\\Lambda_{5}", "score": 0.86, "index": 73}, {"type": "text", "coordinates": [118, 402, 135, 420], "content": " + ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [135, 403, 164, 417], "content": "(2\\Lambda_{1})", "score": 0.87, "index": 75}, {"type": "text", "coordinates": [165, 402, 210, 420], "content": " for any ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [211, 403, 244, 416], "content": "k\\geq2", "score": 0.91, "index": 77}, {"type": "text", "coordinates": [244, 402, 311, 420], "content": ", but equals ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [311, 403, 326, 416], "content": "\\Lambda_{5}", "score": 0.88, "index": 79}, {"type": "text", "coordinates": [326, 402, 344, 420], "content": " at ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [344, 403, 378, 415], "content": "k\\,=\\,1", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [378, 402, 542, 420], "content": ". A similar convention is used", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [70, 418, 541, 433], "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [71, 433, 339, 447], "content": "equal the numerical value of the fusion coefficient).", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [90, 450, 329, 475], "content": "Theorem 3.E6. The fusion-symmetries of ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [330, 450, 353, 468], "content": "E_{6}^{(1)}", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [353, 450, 377, 475], "content": "are ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [378, 452, 414, 468], "content": "C^{i}\\,\\pi[a]", "score": 0.9, "index": 88}, {"type": "text", "coordinates": [415, 450, 465, 475], "content": ", for any ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [465, 453, 516, 468], "content": "i\\in\\{0,1\\}", "score": 0.92, "index": 90}, {"type": "text", "coordinates": [516, 450, 542, 475], "content": " and", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [71, 467, 93, 485], "content": "any ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [94, 468, 156, 482], "content": "a\\in\\{0,1,2\\}", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [156, 467, 212, 485], "content": " for which ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [212, 468, 248, 482], "content": "a k\\not\\equiv1", "score": 0.74, "index": 95}, {"type": "text", "coordinates": [249, 467, 299, 485], "content": " (mod 3).", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [68, 491, 186, 512], "content": "3.6. The algebra E7(1)", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [94, 514, 146, 529], "content": "A weight", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [147, 514, 156, 526], "content": "\\lambda", "score": 0.8, "index": 99}, {"type": "text", "coordinates": [156, 514, 174, 529], "content": " in ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [174, 514, 191, 528], "content": "P_{+}", "score": 0.89, "index": 101}, {"type": "text", "coordinates": [191, 514, 239, 529], "content": " satisfies ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [239, 514, 511, 527], "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "score": 0.88, "index": 103}, {"type": "text", "coordinates": [512, 514, 542, 529], "content": ", and", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [71, 528, 132, 541], "content": "\\kappa\\,=\\,k\\,+\\,18", "score": 0.91, "index": 105}, {"type": "text", "coordinates": [132, 529, 481, 545], "content": ". The charge-conjugation is trivial, but there is a simple-current ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [482, 531, 490, 540], "content": "J", "score": 0.84, "index": 107}, {"type": "text", "coordinates": [491, 529, 541, 545], "content": " given by", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [71, 542, 216, 557], "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "score": 0.91, "index": 109}, {"type": "text", "coordinates": [216, 543, 258, 559], "content": ". It has ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [258, 542, 388, 557], "content": "Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2", "score": 0.92, "index": 111}, {"type": "text", "coordinates": [388, 543, 392, 559], "content": ".", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [96, 558, 465, 572], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [120, 576, 135, 590], "content": "\\Lambda_{6}", "score": 0.87, "index": 114}, {"type": "text", "coordinates": [136, 575, 151, 592], "content": " \u00d7", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [152, 575, 202, 591], "content": "\\Lambda_{6}=(0)_{1}", "score": 0.93, "index": 116}, {"type": "text", "coordinates": [203, 575, 219, 592], "content": " + ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [220, 576, 249, 591], "content": "(\\Lambda_{1})_{2}", "score": 0.92, "index": 118}, {"type": "text", "coordinates": [249, 575, 266, 592], "content": " + ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [266, 575, 295, 591], "content": "(\\Lambda_{5})_{2}", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [296, 575, 313, 592], "content": " + ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [313, 575, 348, 591], "content": "(2\\Lambda_{6})_{2}", "score": 0.87, "index": 122}, {"type": "inline_equation", "coordinates": [120, 594, 135, 608], "content": "\\Lambda_{1}", "score": 0.9, "index": 123}, {"type": "text", "coordinates": [136, 594, 151, 610], "content": " \u00d7", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [152, 594, 210, 609], "content": "\\Lambda_{6}=(\\Lambda_{6})_{2}", "score": 0.93, "index": 125}, {"type": "text", "coordinates": [210, 594, 227, 610], "content": " + ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [227, 594, 257, 609], "content": "(\\Lambda_{7})_{2}", "score": 0.92, "index": 127}, {"type": "text", "coordinates": [257, 594, 273, 610], "content": " + ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [274, 594, 332, 609], "content": "(\\Lambda_{1}+\\Lambda_{6})_{3}", "score": 0.9, "index": 129}, {"type": "inline_equation", "coordinates": [120, 612, 135, 626], "content": "\\Lambda_{5}", "score": 0.89, "index": 130}, {"type": "text", "coordinates": [136, 611, 151, 627], "content": "\u00d7", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [152, 612, 210, 627], "content": "\\Lambda_{6}=(\\Lambda_{4})_{3}", "score": 0.94, "index": 132}, {"type": "text", "coordinates": [210, 611, 227, 627], "content": " + ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [227, 612, 257, 627], "content": "(\\Lambda_{6})_{2}", "score": 0.92, "index": 134}, {"type": "text", "coordinates": [257, 611, 273, 627], "content": " + ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [274, 612, 303, 627], "content": "(\\Lambda_{7})_{2}", "score": 0.93, "index": 136}, {"type": "text", "coordinates": [303, 611, 320, 627], "content": " + ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [321, 612, 378, 627], "content": "(\\Lambda_{1}+\\Lambda_{6})_{3}", "score": 0.92, "index": 138}, {"type": "text", "coordinates": [379, 611, 395, 627], "content": " + ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [395, 612, 453, 627], "content": "(\\Lambda_{5}+\\Lambda_{6})_{3}", "score": 0.86, "index": 140}, {"type": "inline_equation", "coordinates": [105, 630, 120, 644], "content": "\\Lambda_{6}", "score": 0.86, "index": 141}, {"type": "text", "coordinates": [120, 630, 137, 646], "content": " \u00d7 ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [137, 630, 210, 645], "content": 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698], "content": "\\Lambda_{6}", "caption": ""}, {"type": "inline", "coordinates": [115, 683, 210, 698], "content": "(\\Lambda_{5}+\\Lambda_{6})=(\\Lambda_{5})_{3}", "caption": ""}, {"type": "inline", "coordinates": [227, 683, 263, 698], "content": "(2\\Lambda_{5})_{4}", "caption": ""}, {"type": "inline", "coordinates": [280, 683, 315, 698], "content": "(2\\Lambda_{6})_{3}", "caption": ""}, {"type": "inline", "coordinates": [332, 683, 390, 698], "content": "(\\Lambda_{6}+\\Lambda_{7})_{3}", "caption": ""}, {"type": "inline", "coordinates": [407, 683, 465, 698], "content": "(\\Lambda_{1}+\\Lambda_{5})_{4}", "caption": ""}, {"type": "inline", "coordinates": [199, 701, 257, 716], "content": "(\\Lambda_{4}+\\Lambda_{6})_{4}", "caption": ""}, {"type": "inline", "coordinates": [274, 701, 338, 716], "content": "(\\Lambda_{1}+2\\Lambda_{6})_{4}", "caption": ""}, {"type": "inline", "coordinates": [355, 701, 419, 716], "content": "(\\Lambda_{5}+2\\Lambda_{6})_{4}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 12}, {"type": "text", "text": "$\\pi_{v}$ here refers to the simple-current automorphism $\\pi[2]$ or 10 00 ], for r odd/even. When $k\\,=\\,1$ , $A(D_{e v e n,1})\\cong{\\mathfrak{S}}_{3}$ , corresponding to any permutation of $\\Lambda_{1},\\Lambda_{r-1},\\Lambda_{r}$ , and $A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}$ . When $r\\,>\\,4$ , ${\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}$ or $\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}$ for $r$ even/odd. $A(D_{4,2})$ has 24 elements, and any element can be written uniquely as $C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]$ ", "page_idx": 12}, {"type": "text", "text": "3.5. The algebra $E_{6}^{(1)}$ ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}$ and $\\kappa=k\\!+\\!12$ . The charge-conjugation acts as $C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})$ . The order 3 simple-current $J$ is given by $J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})$ with $Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3$ . ", "page_idx": 12}, {"type": "text", "text": "The fusion products we need can be derived from [29] using (2.4): ", "page_idx": 12}, {"type": "equation", "text": "$$\n(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "where the outer subscript on any summand denotes the smallest level where that summand appears (it will also appear at all larger levels). So for example $\\Lambda_{1}\\boxtimes\\Lambda_{1}$ equals $\\Lambda_{2}$ + $\\Lambda_{5}$ + $(2\\Lambda_{1})$ for any $k\\geq2$ , but equals $\\Lambda_{5}$ at $k\\,=\\,1$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient). ", "page_idx": 12}, {"type": "text", "text": "Theorem 3.E6. The fusion-symmetries of $E_{6}^{(1)}$ are $C^{i}\\,\\pi[a]$ , for any $i\\in\\{0,1\\}$ and any $a\\in\\{0,1,2\\}$ for which $a k\\not\\equiv1$ (mod 3). ", "page_idx": 12}, {"type": "text", "text": "3.6. The algebra E7(1) ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}$ , and $\\kappa\\,=\\,k\\,+\\,18$ . The charge-conjugation is trivial, but there is a simple-current $J$ given by $J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})$ . It has $Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2$ . ", "page_idx": 12}, {"type": "text", "text": "The only fusion products we need can be obtained from [29] and (2.4): ", "page_idx": 12}, {"type": "text", "text": "$\\Lambda_{6}$ \u00d7 $\\Lambda_{6}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{5})_{2}$ + $(2\\Lambda_{6})_{2}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ $\\Lambda_{5}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{4})_{3}$ + $(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{6}$ \u00d7 $(2\\Lambda_{6})=(\\Lambda_{6})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(3\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{4}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{5})_{3}$ + $(\\Lambda_{1}+\\Lambda_{5})_{4}$ + $(\\Lambda_{4}+\\Lambda_{6})_{4}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ $\\Lambda_{6}$ \u00d7 $\\Lambda_{7}=(\\Lambda_{1})_{2}$ + 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The order 3 simple-current ", "type": "text"}, {"bbox": [531, 256, 540, 265], "score": 0.87, "content": "J", "type": "inline_equation", "height": 9, "width": 9}], "index": 9}, {"bbox": [69, 267, 490, 284], "spans": [{"bbox": [69, 268, 131, 284], "score": 1.0, "content": "is given by ", "type": "text"}, {"bbox": [131, 268, 290, 282], "score": 0.92, "content": "J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})", "type": "inline_equation", "height": 14, "width": 159}, {"bbox": [290, 268, 319, 284], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [320, 267, 487, 282], "score": 0.91, "content": "Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3", "type": "inline_equation", "height": 15, "width": 167}, {"bbox": [487, 268, 490, 284], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [96, 281, 443, 295], "lines": [{"bbox": [95, 282, 440, 297], "spans": [{"bbox": [95, 282, 440, 297], "score": 1.0, "content": "The fusion products we need can be derived from [29] using (2.4):", "type": "text"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [282, 354, 416, 370], "lines": [{"bbox": [282, 354, 416, 370], "spans": [{"bbox": [282, 354, 416, 370], "score": 0.38, "content": "(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [70, 372, 541, 445], "lines": [{"bbox": [70, 374, 540, 390], "spans": [{"bbox": [70, 374, 540, 390], "score": 1.0, "content": "where the outer subscript on any summand denotes the smallest level where that sum-", "type": "text"}], "index": 13}, {"bbox": [70, 388, 541, 403], "spans": [{"bbox": [70, 389, 455, 403], "score": 1.0, "content": "mand appears (it will also appear at all larger levels). So for example ", "type": "text"}, {"bbox": [456, 388, 502, 402], "score": 0.36, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [503, 389, 541, 403], "score": 1.0, "content": " equals", "type": "text"}], "index": 14}, {"bbox": [71, 402, 542, 420], "spans": [{"bbox": [71, 404, 85, 416], "score": 0.86, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [86, 402, 102, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [102, 403, 117, 416], "score": 0.86, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [118, 402, 135, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [135, 403, 164, 417], "score": 0.87, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [165, 402, 210, 420], "score": 1.0, "content": " for any ", "type": "text"}, {"bbox": [211, 403, 244, 416], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [244, 402, 311, 420], "score": 1.0, "content": ", but equals ", "type": "text"}, {"bbox": [311, 403, 326, 416], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [326, 402, 344, 420], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [344, 403, 378, 415], "score": 0.91, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [378, 402, 542, 420], "score": 1.0, "content": ". A similar convention is used", "type": "text"}], "index": 15}, {"bbox": [70, 418, 541, 433], "spans": [{"bbox": [70, 418, 541, 433], "score": 1.0, "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "type": "text"}], "index": 16}, {"bbox": [71, 433, 339, 447], "spans": [{"bbox": [71, 433, 339, 447], "score": 1.0, "content": "equal the numerical value of the fusion coefficient).", "type": "text"}], "index": 17}], "index": 15}, {"type": "text", "bbox": [71, 450, 542, 482], "lines": [{"bbox": [90, 450, 542, 475], "spans": [{"bbox": [90, 450, 329, 475], "score": 1.0, "content": "Theorem 3.E6. The fusion-symmetries of ", "type": "text"}, {"bbox": [330, 450, 353, 468], "score": 0.92, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [353, 450, 377, 475], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [378, 452, 414, 468], "score": 0.9, "content": "C^{i}\\,\\pi[a]", "type": "inline_equation", "height": 16, "width": 36}, {"bbox": [415, 450, 465, 475], "score": 1.0, "content": ", for any ", "type": "text"}, {"bbox": [465, 453, 516, 468], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 51}, {"bbox": [516, 450, 542, 475], "score": 1.0, "content": " and", "type": "text"}], "index": 18}, {"bbox": [71, 467, 299, 485], "spans": [{"bbox": [71, 467, 93, 485], "score": 1.0, "content": "any ", "type": "text"}, {"bbox": [94, 468, 156, 482], "score": 0.93, "content": "a\\in\\{0,1,2\\}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [156, 467, 212, 485], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [212, 468, 248, 482], "score": 0.74, "content": "a k\\not\\equiv1", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [249, 467, 299, 485], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "title", "bbox": [71, 491, 183, 507], "lines": [{"bbox": [68, 491, 186, 512], "spans": [{"bbox": [68, 491, 186, 512], "score": 1.0, "content": "3.6. The algebra E7(1)", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 512, 542, 556], "lines": [{"bbox": [94, 514, 542, 529], "spans": [{"bbox": [94, 514, 146, 529], "score": 1.0, "content": "A weight", "type": "text"}, {"bbox": [147, 514, 156, 526], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [156, 514, 174, 529], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [174, 514, 191, 528], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [191, 514, 239, 529], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [239, 514, 511, 527], "score": 0.88, "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "type": "inline_equation", "height": 13, "width": 272}, {"bbox": [512, 514, 542, 529], "score": 1.0, "content": ", and", "type": "text"}], "index": 21}, {"bbox": [71, 528, 541, 545], "spans": [{"bbox": [71, 528, 132, 541], "score": 0.91, "content": "\\kappa\\,=\\,k\\,+\\,18", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [132, 529, 481, 545], "score": 1.0, "content": ". The charge-conjugation is trivial, but there is a simple-current ", "type": "text"}, {"bbox": [482, 531, 490, 540], "score": 0.84, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [491, 529, 541, 545], "score": 1.0, "content": " given by", "type": "text"}], "index": 22}, {"bbox": [71, 542, 392, 559], "spans": [{"bbox": [71, 542, 216, 557], "score": 0.91, "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "type": "inline_equation", "height": 15, "width": 145}, {"bbox": [216, 543, 258, 559], "score": 1.0, "content": ". 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The", "type": "text"}], "index": 8}, {"bbox": [72, 253, 540, 269], "spans": [{"bbox": [72, 253, 213, 269], "score": 1.0, "content": "charge-conjugation acts as ", "type": "text"}, {"bbox": [213, 253, 375, 267], "score": 0.89, "content": "C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})", "type": "inline_equation", "height": 14, "width": 162}, {"bbox": [376, 253, 531, 269], "score": 1.0, "content": ". 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A similar convention is used", "type": "text"}], "index": 15}, {"bbox": [70, 418, 541, 433], "spans": [{"bbox": [70, 418, 541, 433], "score": 1.0, "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "type": "text"}], "index": 16}, {"bbox": [71, 433, 339, 447], "spans": [{"bbox": [71, 433, 339, 447], "score": 1.0, "content": "equal the numerical value of the fusion coefficient).", "type": "text"}], "index": 17}], "index": 15, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [70, 374, 542, 447]}, {"type": "text", "bbox": [71, 450, 542, 482], "lines": [{"bbox": [90, 450, 542, 475], "spans": [{"bbox": [90, 450, 329, 475], "score": 1.0, "content": "Theorem 3.E6. The fusion-symmetries of ", "type": "text"}, {"bbox": [330, 450, 353, 468], "score": 0.92, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [353, 450, 377, 475], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [378, 452, 414, 468], "score": 0.9, "content": "C^{i}\\,\\pi[a]", "type": "inline_equation", "height": 16, "width": 36}, {"bbox": [415, 450, 465, 475], "score": 1.0, "content": ", for any ", "type": "text"}, {"bbox": [465, 453, 516, 468], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 51}, {"bbox": [516, 450, 542, 475], "score": 1.0, "content": " and", "type": "text"}], "index": 18}, {"bbox": [71, 467, 299, 485], "spans": [{"bbox": [71, 467, 93, 485], "score": 1.0, "content": "any ", "type": "text"}, {"bbox": [94, 468, 156, 482], "score": 0.93, "content": "a\\in\\{0,1,2\\}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [156, 467, 212, 485], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [212, 468, 248, 482], "score": 0.74, "content": "a k\\not\\equiv1", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [249, 467, 299, 485], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [71, 450, 542, 485]}, {"type": "title", "bbox": [71, 491, 183, 507], "lines": [{"bbox": [68, 491, 186, 512], "spans": [{"bbox": [68, 491, 186, 512], "score": 1.0, "content": "3.6. The algebra E7(1)", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 512, 542, 556], "lines": [{"bbox": [94, 514, 542, 529], "spans": [{"bbox": [94, 514, 146, 529], "score": 1.0, "content": "A weight", "type": "text"}, {"bbox": [147, 514, 156, 526], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [156, 514, 174, 529], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [174, 514, 191, 528], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [191, 514, 239, 529], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [239, 514, 511, 527], "score": 0.88, "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "type": "inline_equation", "height": 13, "width": 272}, {"bbox": [512, 514, 542, 529], "score": 1.0, "content": ", and", "type": "text"}], "index": 21}, {"bbox": [71, 528, 541, 545], "spans": [{"bbox": [71, 528, 132, 541], "score": 0.91, "content": "\\kappa\\,=\\,k\\,+\\,18", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [132, 529, 481, 545], "score": 1.0, "content": ". The charge-conjugation is trivial, but there is a simple-current ", "type": "text"}, {"bbox": [482, 531, 490, 540], "score": 0.84, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [491, 529, 541, 545], "score": 1.0, "content": " given by", "type": "text"}], "index": 22}, {"bbox": [71, 542, 392, 559], "spans": [{"bbox": [71, 542, 216, 557], "score": 0.91, "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "type": "inline_equation", "height": 15, "width": 145}, {"bbox": [216, 543, 258, 559], "score": 1.0, "content": ". It has ", "type": "text"}, {"bbox": [258, 542, 388, 557], "score": 0.92, "content": "Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2", "type": "inline_equation", "height": 15, "width": 130}, {"bbox": [388, 543, 392, 559], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [71, 514, 542, 559]}, {"type": "text", "bbox": [94, 556, 468, 570], "lines": [{"bbox": [96, 558, 465, 572], "spans": [{"bbox": [96, 558, 465, 572], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [96, 558, 465, 572]}, {"type": "text", "bbox": [82, 573, 530, 716], "lines": [{"bbox": [120, 575, 348, 592], "spans": [{"bbox": [120, 576, 135, 590], "score": 0.87, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [136, 575, 151, 592], 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0002044v1	8	Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras $$X_{r}^{(1)}$$ , for simple $$X_{r}$$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $$\pi[a]$$ and Galois automorphism $$\pi\{\ell\}$$ defined in §2.3, and the notation $$\kappa=k\!+\!h^{\vee}$$ . It will be convenient to write $$X{_{r,k}}^{,}$$ for $$\cdot X_{r}^{(1)}$$ and level $$k'$$ . We write $$_S$$ for the group of symmetries of the extended Dynkin diagram. # 3.1. The algebra $$A_{r}^{(1)}$$ , $$r\geq1$$ Define $$\overline{r}\,=\,r\,+\,1$$ and $$\;n\;=\;k\,+\,{\overline{{r}}}$$ . The level $$k$$ highest weights of $$A_{r}^{(1)}$$ constitute the set $$P_{+}$$ of $$\overline{r}$$ -tuples $$\lambda\,=\,(\lambda_{0},.\,.\,.\,,\lambda_{r})$$ of non-negative integers obeying $$\textstyle\sum_{i=0}^{r}\lambda_{i}\;=\;k$$ . The Dynkin diagram symmetries form the dihedral group $$\boldsymbol{S}\;=\;\mathfrak{D}_{r+1}$$ ; it is generated by the charge-conjugation $$C$$ and simple-current $$J$$ given by $$C\lambda\,=\,(\lambda_{0},\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1})$$ and $$J\lambda=\left(\lambda_{r},\lambda_{0},\lambda_{1},.\dots,\lambda_{r-1}\right)$$ , with $$Q_{J^{a}}(\lambda)=a\,t(\lambda)/\overline{{r}}$$ for $$\begin{array}{r}{t(\lambda)\overset{\mathrm{def}}{=}\sum_{j=1}^{r}j\lambda_{j}}\end{array}$$ . Note that $$C=i d$$ . for $${A}_{1}^{(1)}$$ . The Kac-Peterson relation (2.1b) for $$A_{r,k}$$ takes the form where $$s_{(\lambda)}\big(x_{1},\ldots,x_{r+1}\big)$$ is the Schur polynomial (see e.g. [27]) corresponding to the parti- tion $$(\lambda(1),\ldots,\lambda(\overline{{r}}))$$ , and where $$\textstyle\nu(\ell)=\sum_{i=\ell}^{r}\nu_{i}$$ for any weight $$\nu$$ . In other words, $$S_{\lambda\mu}/S_{0\mu}$$ is the Schur polynomial corresponding to $$\lambda$$ , evaluated at roots of 1 determined by $$\mu$$ . The fusion (derived from the Pieri rule and (2.4)) valid for $$k\geq2$$ and $$1\leq\ell<r$$ , will be useful. There are no exceptional fusion-symmetries for $$A_{r}^{(1)}$$ : Theorem 3.A. The fusion-symmetries for $$A_{r}^{(1)}$$ level $$k$$ are $$C^{i}\pi[a]$$ , for $$i\in\{0,1\}$$ and any integer $$0\leq a\leq r$$ for which $$1+k a$$ is coprime to $$r+1$$ . To avoid redundancies in the Theorem, for $$r\,=\,1$$ or $$k\,=\,1$$ take $$i\,=\,0$$ only. If we write $${\overline{{r}}}\,=\,r^{\prime}r^{\prime\prime}$$ , where $$r^{\prime}$$ is coprime to $$k$$ and $$r^{\prime\prime}\|k^{\infty}$$ , then the number of simple-current automorphisms will exactly equal $$r^{\prime\prime}{\cdot}\varphi(r^{\prime})$$ , where $$\varphi$$ is the Euler totient. The $$\pi[a]$$ commute with each other, and with $$C$$ . For example, for $$A_{1,k}$$ when $$k$$ is odd, there is no nontrivial fusion-symmetry. When $$k$$ is even, there is exactly one, sending $$\lambda=\lambda_{1}\Lambda_{1}$$ to $$\lambda$$ (for $$\lambda_{1}$$ even) or $$J\lambda=(k-\lambda_{1})\Lambda_{1}$$ (for $$\lambda_{1}$$ odd). For $$A_{2,k}$$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $$k$$ .	<p>Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras $$X_{r}^{(1)}$$ , for simple $$X_{r}$$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $$\pi[a]$$ and Galois automorphism $$\pi\{\ell\}$$ defined in §2.3, and the notation $$\kappa=k\!+\!h^{\vee}$$ . It will be convenient to write $$X{_{r,k}}^{,}$$ for $$\cdot X_{r}^{(1)}$$ and level $$k'$$ . We write $$_S$$ for the group of symmetries of the extended Dynkin diagram.</p> <h1>3.1. The algebra $$A_{r}^{(1)}$$ , $$r\geq1$$</h1> <p>Define $$\overline{r}\,=\,r\,+\,1$$ and $$\;n\;=\;k\,+\,{\overline{{r}}}$$ . The level $$k$$ highest weights of $$A_{r}^{(1)}$$ constitute the set $$P_{+}$$ of $$\overline{r}$$ -tuples $$\lambda\,=\,(\lambda_{0},.\,.\,.\,,\lambda_{r})$$ of non-negative integers obeying $$\textstyle\sum_{i=0}^{r}\lambda_{i}\;=\;k$$ . The Dynkin diagram symmetries form the dihedral group $$\boldsymbol{S}\;=\;\mathfrak{D}_{r+1}$$ ; it is generated by the charge-conjugation $$C$$ and simple-current $$J$$ given by $$C\lambda\,=\,(\lambda_{0},\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1})$$ and $$J\lambda=\left(\lambda_{r},\lambda_{0},\lambda_{1},.\dots,\lambda_{r-1}\right)$$ , with $$Q_{J^{a}}(\lambda)=a\,t(\lambda)/\overline{{r}}$$ for $$\begin{array}{r}{t(\lambda)\overset{\mathrm{def}}{=}\sum_{j=1}^{r}j\lambda_{j}}\end{array}$$ . Note that $$C=i d$$ . for $${A}_{1}^{(1)}$$ .</p> <p>The Kac-Peterson relation (2.1b) for $$A_{r,k}$$ takes the form</p> <p>where $$s_{(\lambda)}\big(x_{1},\ldots,x_{r+1}\big)$$ is the Schur polynomial (see e.g. [27]) corresponding to the parti- tion $$(\lambda(1),\ldots,\lambda(\overline{{r}}))$$ , and where $$\textstyle\nu(\ell)=\sum_{i=\ell}^{r}\nu_{i}$$ for any weight $$\nu$$ . In other words, $$S_{\lambda\mu}/S_{0\mu}$$ is the Schur polynomial corresponding to $$\lambda$$ , evaluated at roots of 1 determined by $$\mu$$ .</p> <p>The fusion (derived from the Pieri rule and (2.4))</p> <p>valid for $$k\geq2$$ and $$1\leq\ell<r$$ , will be useful.</p> <p>There are no exceptional fusion-symmetries for $$A_{r}^{(1)}$$ :</p> <p>Theorem 3.A. The fusion-symmetries for $$A_{r}^{(1)}$$ level $$k$$ are $$C^{i}\pi[a]$$ , for $$i\in\{0,1\}$$ and any integer $$0\leq a\leq r$$ for which $$1+k a$$ is coprime to $$r+1$$ .</p> <p>To avoid redundancies in the Theorem, for $$r\,=\,1$$ or $$k\,=\,1$$ take $$i\,=\,0$$ only. If we write $${\overline{{r}}}\,=\,r^{\prime}r^{\prime\prime}$$ , where $$r^{\prime}$$ is coprime to $$k$$ and $$r^{\prime\prime}\|k^{\infty}$$ , then the number of simple-current automorphisms will exactly equal $$r^{\prime\prime}{\cdot}\varphi(r^{\prime})$$ , where $$\varphi$$ is the Euler totient. The $$\pi[a]$$ commute with each other, and with $$C$$ .</p> <p>For example, for $$A_{1,k}$$ when $$k$$ is odd, there is no nontrivial fusion-symmetry. When $$k$$ is even, there is exactly one, sending $$\lambda=\lambda_{1}\Lambda_{1}$$ to $$\lambda$$ (for $$\lambda_{1}$$ even) or $$J\lambda=(k-\lambda_{1})\Lambda_{1}$$ (for $$\lambda_{1}$$ odd). For $$A_{2,k}$$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $$k$$ .</p>	[{"type": "text", "coordinates": [70, 99, 542, 191], "content": "Our main task in this paper is to find and construct all fusion-symmetries for the\naffine algebras $$X_{r}^{(1)}$$ , for simple $$X_{r}$$ . In this section we state the results, and in the next\nsection we prove the completeness of our lists. Recall the simple-current automorphism\n$$\\pi[a]$$ and Galois automorphism $$\\pi\\{\\ell\\}$$ defined in \u00a72.3, and the notation $$\\kappa=k\\!+\\!h^{\\vee}$$ . It will be\nconvenient to write $$X{_{r,k}}^{,}$$ for $$\\cdot X_{r}^{(1)}$$ and level $$k'$$ . We write $$_S$$ for the group of symmetries\nof the extended Dynkin diagram.", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [71, 203, 218, 221], "content": "3.1. The algebra $$A_{r}^{(1)}$$ , $$r\\geq1$$", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [69, 228, 541, 323], "content": "Define $$\\overline{r}\\,=\\,r\\,+\\,1$$ and $$\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}$$ . The level $$k$$ highest weights of $$A_{r}^{(1)}$$ constitute\nthe set $$P_{+}$$ of $$\\overline{r}$$ -tuples $$\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})$$ of non-negative integers obeying $$\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k$$ .\nThe Dynkin diagram symmetries form the dihedral group $$\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}$$ ; it is generated\nby the charge-conjugation $$C$$ and simple-current $$J$$ given by $$C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})$$\nand $$J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)$$ , with $$Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}$$ for $$\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}$$ . Note that\n$$C=i d$$ . for $${A}_{1}^{(1)}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [95, 324, 394, 339], "content": "The Kac-Peterson relation (2.1b) for $$A_{r,k}$$ takes the form", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [87, 352, 487, 383], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 395, 540, 439], "content": "where $$s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)$$ is the Schur polynomial (see e.g. [27]) corresponding to the parti-\ntion $$(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))$$ , and where $$\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}$$ for any weight $$\\nu$$ . In other words, $$S_{\\lambda\\mu}/S_{0\\mu}$$\nis the Schur polynomial corresponding to $$\\lambda$$ , evaluated at roots of 1 determined by $$\\mu$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [94, 439, 357, 453], "content": "The fusion (derived from the Pieri rule and (2.4))", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [223, 467, 386, 484], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [71, 496, 304, 510], "content": "valid for $$k\\geq2$$ and $$1\\leq\\ell<r$$ , will be useful.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [95, 512, 371, 527], "content": "There are no exceptional fusion-symmetries for $$A_{r}^{(1)}$$ :", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [70, 533, 542, 565], "content": "Theorem 3.A. The fusion-symmetries for $$A_{r}^{(1)}$$ level $$k$$ are $$C^{i}\\pi[a]$$ , for $$i\\in\\{0,1\\}$$ and\nany integer $$0\\leq a\\leq r$$ for which $$1+k a$$ is coprime to $$r+1$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 571, 541, 628], "content": "To avoid redundancies in the Theorem, for $$r\\,=\\,1$$ or $$k\\,=\\,1$$ take $$i\\,=\\,0$$ only. If we\nwrite $${\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}$$ , where $$r^{\\prime}$$ is coprime to $$k$$ and $$r^{\\prime\\prime}\|k^{\\infty}$$ , then the number of simple-current\nautomorphisms will exactly equal $$r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})$$ , where $$\\varphi$$ is the Euler totient. The $$\\pi[a]$$ commute\nwith each other, and with $$C$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [70, 630, 541, 686], "content": "For example, for $$A_{1,k}$$ when $$k$$ is odd, there is no nontrivial fusion-symmetry. When $$k$$\nis even, there is exactly one, sending $$\\lambda=\\lambda_{1}\\Lambda_{1}$$ to $$\\lambda$$ (for $$\\lambda_{1}$$ even) or $$J\\lambda=(k-\\lambda_{1})\\Lambda_{1}$$ (for\n$$\\lambda_{1}$$ odd). For $$A_{2,k}$$ , there are either six or four fusion-symmetries, depending on whether\nor not 3 divides $$k$$ .", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [95, 102, 541, 117], "content": "Our main task in this paper is to find and construct all fusion-symmetries for the", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [69, 113, 151, 135], "content": "affine algebras ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [151, 116, 174, 130], "content": "X_{r}^{(1)}", "score": 0.94, "index": 3}, {"type": "text", "coordinates": [175, 113, 239, 135], "content": ", for simple ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [239, 120, 255, 131], "content": "X_{r}", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [255, 113, 544, 135], "content": ". In this section we state the results, and in the next", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [70, 132, 542, 148], "content": "section we prove the completeness of our lists. Recall the simple-current automorphism", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [71, 148, 92, 161], "content": "\\pi[a]", "score": 0.92, "index": 8}, {"type": "text", "coordinates": [92, 146, 231, 162], "content": " and Galois automorphism ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [231, 148, 256, 161], "content": "\\pi\\{\\ell\\}", "score": 0.91, "index": 10}, {"type": "text", "coordinates": [257, 146, 429, 162], "content": " defined in \u00a72.3, and the notation ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [429, 148, 484, 159], "content": "\\kappa=k\\!+\\!h^{\\vee}", "score": 0.94, "index": 12}, {"type": "text", "coordinates": [485, 146, 541, 162], "content": ". It will be", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [68, 163, 178, 178], "content": "convenient to write ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [179, 164, 206, 178], "content": "X{_{r,k}}^{,}", "score": 0.86, "index": 15}, {"type": "text", "coordinates": [206, 163, 230, 178], "content": " for ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [231, 161, 255, 176], "content": "\\cdot X_{r}^{(1)}", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [257, 163, 310, 178], "content": "and level ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [310, 165, 320, 174], "content": "k'", "score": 0.62, "index": 19}, {"type": "text", "coordinates": [321, 163, 380, 178], "content": ". We write ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [381, 166, 389, 174], "content": "_S", "score": 0.91, "index": 21}, {"type": "text", "coordinates": [389, 163, 541, 178], "content": " for the group of symmetries", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [70, 177, 245, 192], "content": "of the extended Dynkin diagram.", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [69, 202, 160, 225], "content": "3.1. The algebra ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [161, 205, 183, 221], "content": "A_{r}^{(1)}", "score": 0.48, "index": 25}, {"type": "text", "coordinates": [183, 202, 189, 225], "content": ", ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [190, 208, 218, 221], "content": "r\\geq1", "score": 0.48, "index": 27}, {"type": "text", "coordinates": [93, 228, 133, 247], "content": "Define ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [133, 232, 190, 243], "content": "\\overline{r}\\,=\\,r\\,+\\,1", "score": 0.89, "index": 29}, {"type": "text", "coordinates": [190, 228, 219, 247], "content": " and ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [219, 233, 277, 243], "content": "\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [278, 228, 345, 247], "content": ". The level ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [345, 234, 352, 243], "content": "k", "score": 0.87, "index": 33}, {"type": "text", "coordinates": [353, 228, 459, 247], "content": " highest weights of ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [460, 228, 482, 244], "content": "A_{r}^{(1)}", "score": 0.91, "index": 35}, {"type": "text", "coordinates": [483, 228, 542, 247], "content": "constitute", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [70, 245, 112, 262], "content": "the set ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [113, 246, 129, 259], "content": "P_{+}", "score": 0.89, "index": 38}, {"type": "text", "coordinates": [129, 245, 148, 262], "content": " of ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [148, 247, 155, 257], "content": "\\overline{r}", "score": 0.75, "index": 40}, {"type": "text", "coordinates": [156, 245, 195, 262], "content": "-tuples ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [195, 246, 284, 259], "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [284, 245, 466, 262], "content": " of non-negative integers obeying", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [467, 245, 537, 261], "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "score": 0.91, "index": 44}, {"type": "text", "coordinates": [537, 245, 541, 262], "content": ".", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [70, 259, 392, 276], "content": "The Dynkin diagram symmetries form the dihedral group ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [392, 261, 451, 274], "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "score": 0.91, "index": 47}, {"type": "text", "coordinates": [451, 259, 542, 276], "content": "; it is generated", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [70, 274, 213, 290], "content": "by the charge-conjugation ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [214, 275, 224, 285], "content": "C", "score": 0.83, "index": 50}, {"type": "text", "coordinates": [225, 274, 332, 290], "content": " and simple-current ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [333, 276, 341, 285], "content": "J", "score": 0.88, "index": 52}, {"type": "text", "coordinates": [341, 274, 395, 290], "content": " given by ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [395, 274, 540, 288], "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "score": 0.89, "index": 54}, {"type": "text", "coordinates": [69, 291, 94, 309], "content": "and ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [95, 292, 235, 306], "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [235, 291, 270, 309], "content": ", with ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [270, 293, 366, 306], "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [367, 291, 388, 309], "content": " for ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [389, 290, 478, 308], "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "score": 0.92, "index": 60}, {"type": "text", "coordinates": [479, 291, 542, 309], "content": ". Note that", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [71, 313, 108, 322], "content": "C=i d", "score": 0.88, "index": 62}, {"type": "text", "coordinates": [108, 310, 132, 325], "content": ". for ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [132, 309, 154, 325], "content": "{A}_{1}^{(1)}", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [155, 310, 159, 325], "content": ".", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [94, 325, 290, 341], "content": "The Kac-Peterson relation (2.1b) for ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [290, 328, 313, 340], "content": "A_{r,k}", "score": 0.92, "index": 67}, {"type": "text", "coordinates": [313, 325, 393, 341], "content": " takes the form", "score": 1.0, "index": 68}, {"type": "interline_equation", "coordinates": [87, 352, 487, 383], "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "score": 0.85, "index": 69}, {"type": "text", "coordinates": [71, 397, 105, 414], "content": "where ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [105, 399, 197, 412], "content": "s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [198, 397, 540, 414], "content": " is the Schur polynomial (see e.g. [27]) corresponding to the parti-", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [69, 411, 96, 430], "content": "tion ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [96, 413, 176, 426], "content": "(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))", "score": 0.9, "index": 74}, {"type": "text", "coordinates": [176, 411, 239, 430], "content": ", and where ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [239, 411, 317, 427], "content": "\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}", "score": 0.94, "index": 76}, {"type": "text", "coordinates": [317, 411, 397, 430], "content": " for any weight ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [397, 415, 405, 423], "content": "\\nu", "score": 0.72, "index": 78}, {"type": "text", "coordinates": [405, 411, 495, 430], "content": ". In other words, ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [495, 414, 539, 426], "content": "S_{\\lambda\\mu}/S_{0\\mu}", "score": 0.94, "index": 80}, {"type": "text", "coordinates": [69, 425, 289, 443], "content": "is the Schur polynomial corresponding to", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [290, 428, 298, 438], "content": "\\lambda", "score": 0.81, "index": 82}, {"type": "text", "coordinates": [298, 425, 505, 443], "content": ", evaluated at roots of 1 determined by ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [506, 432, 513, 440], "content": "\\mu", "score": 0.89, "index": 84}, {"type": "text", "coordinates": [514, 425, 519, 443], "content": ".", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [96, 441, 355, 455], "content": "The fusion (derived from the Pieri rule and (2.4))", "score": 1.0, "index": 86}, {"type": "interline_equation", "coordinates": [223, 467, 386, 484], "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "score": 0.55, "index": 87}, {"type": "text", "coordinates": [71, 498, 118, 511], "content": "valid for ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [118, 500, 147, 511], "content": "k\\geq2", "score": 0.92, "index": 89}, {"type": "text", "coordinates": [147, 498, 173, 511], "content": " and ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [174, 499, 223, 511], "content": "1\\leq\\ell<r", "score": 0.9, "index": 91}, {"type": "text", "coordinates": [224, 498, 302, 511], "content": ", will be useful.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [93, 512, 344, 529], "content": "There are no exceptional fusion-symmetries for ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [345, 510, 367, 527], "content": "A_{r}^{(1)}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [367, 512, 372, 529], "content": ":", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [93, 534, 323, 554], "content": "Theorem 3.A. The fusion-symmetries for ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [323, 533, 346, 550], "content": "A_{r}^{(1)}", "score": 0.91, "index": 97}, {"type": "text", "coordinates": [347, 534, 375, 554], "content": "level", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [375, 537, 384, 549], "content": "k", "score": 0.72, "index": 99}, {"type": "text", "coordinates": [384, 534, 407, 554], "content": " are ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [407, 536, 442, 551], "content": "C^{i}\\pi[a]", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [442, 534, 467, 554], "content": ", for ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [467, 537, 517, 551], "content": "i\\in\\{0,1\\}", "score": 0.92, "index": 103}, {"type": "text", "coordinates": [517, 534, 543, 554], "content": " and", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [72, 551, 133, 568], "content": "any integer ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [134, 553, 184, 565], "content": "0\\leq a\\leq r", "score": 0.86, "index": 106}, {"type": "text", "coordinates": [185, 551, 240, 568], "content": " for which ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [240, 552, 275, 564], "content": "1+k a", "score": 0.88, "index": 108}, {"type": "text", "coordinates": [275, 551, 350, 568], "content": " is coprime to ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [350, 552, 378, 564], "content": "r+1", "score": 0.9, "index": 110}, {"type": "text", "coordinates": [378, 551, 383, 568], "content": ".", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [94, 574, 329, 587], "content": "To avoid redundancies in the Theorem, for ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [330, 574, 362, 584], "content": "r\\,=\\,1", "score": 0.89, "index": 113}, {"type": "text", "coordinates": [362, 574, 381, 587], "content": " or ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [381, 573, 414, 585], "content": "k\\,=\\,1", "score": 0.89, "index": 115}, {"type": "text", "coordinates": [415, 574, 444, 587], "content": " take ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [445, 574, 475, 585], "content": "i\\,=\\,0", "score": 0.88, "index": 117}, {"type": "text", "coordinates": [476, 574, 541, 587], "content": " only. If we", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [72, 588, 102, 601], "content": "write ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [102, 589, 149, 599], "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [149, 588, 191, 601], "content": ", where ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [192, 589, 201, 598], "content": "r^{\\prime}", "score": 0.9, "index": 122}, {"type": "text", "coordinates": [201, 588, 280, 601], "content": " is coprime to ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [280, 590, 287, 599], "content": "k", "score": 0.89, "index": 124}, {"type": "text", "coordinates": [288, 588, 315, 601], "content": " and ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [316, 588, 348, 601], "content": "r^{\\prime\\prime}\|k^{\\infty}", "score": 0.92, "index": 126}, {"type": "text", "coordinates": [348, 588, 541, 601], "content": ", then the number of simple-current", "score": 1.0, "index": 127}, {"type": "text", "coordinates": [70, 602, 246, 617], "content": "automorphisms will exactly equal ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [246, 603, 289, 616], "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "score": 0.93, "index": 129}, {"type": "text", "coordinates": [290, 602, 329, 617], "content": ", where ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [329, 605, 338, 615], "content": "\\varphi", "score": 0.81, "index": 131}, {"type": "text", "coordinates": [338, 602, 468, 617], "content": " is the Euler totient. The ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [468, 601, 489, 616], "content": "\\pi[a]", "score": 0.92, "index": 133}, {"type": "text", "coordinates": [489, 602, 542, 617], "content": " commute", "score": 1.0, "index": 134}, {"type": "text", "coordinates": [72, 617, 210, 630], "content": "with each other, and with ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [210, 618, 220, 627], "content": "C", "score": 0.89, "index": 136}, {"type": "text", "coordinates": [220, 617, 224, 630], "content": ".", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [95, 631, 184, 646], "content": "For example, for ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [185, 633, 207, 645], "content": "A_{1,k}", "score": 0.93, "index": 139}, {"type": "text", "coordinates": [207, 631, 241, 646], "content": " when ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [242, 633, 248, 641], "content": "k", "score": 0.89, "index": 141}, {"type": "text", "coordinates": [249, 631, 532, 646], "content": " is odd, there is no nontrivial fusion-symmetry. When ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [532, 633, 540, 642], "content": "k", "score": 0.83, "index": 143}, {"type": "text", "coordinates": [69, 645, 265, 661], "content": "is even, there is exactly one, sending ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [266, 647, 315, 658], "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [315, 645, 333, 661], "content": " to ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [333, 647, 340, 656], "content": "\\lambda", "score": 0.87, "index": 147}, {"type": "text", "coordinates": [340, 645, 366, 661], "content": " (for ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [367, 647, 379, 658], "content": "\\lambda_{1}", "score": 0.9, "index": 149}, {"type": "text", "coordinates": [379, 645, 429, 661], "content": " even) or ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [429, 646, 516, 659], "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "score": 0.93, "index": 151}, {"type": "text", "coordinates": [517, 645, 541, 661], "content": " (for", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [71, 662, 83, 672], "content": "\\lambda_{1}", "score": 0.89, "index": 153}, {"type": "text", "coordinates": [84, 659, 144, 675], "content": " odd). 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task in this paper is to find and construct all fusion-symmetries for the affine algebras $X_{r}^{(1)}$ , for simple $X_{r}$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $\\pi[a]$ and Galois automorphism $\\pi\\{\\ell\\}$ defined in \u00a72.3, and the notation $\\kappa=k\\!+\\!h^{\\vee}$ . It will be convenient to write $X{_{r,k}}^{,}$ for $\\cdot X_{r}^{(1)}$ and level $k'$ . We write $_S$ for the group of symmetries of the extended Dynkin diagram. ", "page_idx": 8}, {"type": "text", "text": "3.1. The algebra $A_{r}^{(1)}$ , $r\\geq1$ ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "Define $\\overline{r}\\,=\\,r\\,+\\,1$ and $\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}$ . The level $k$ highest weights of $A_{r}^{(1)}$ constitute the set $P_{+}$ of $\\overline{r}$ -tuples $\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})$ of non-negative integers obeying $\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k$ . The Dynkin diagram symmetries form the dihedral group $\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}$ ; it is generated by the charge-conjugation $C$ and simple-current $J$ given by $C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})$ and $J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)$ , with $Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}$ for $\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}$ . Note that $C=i d$ . for ${A}_{1}^{(1)}$ . ", "page_idx": 8}, {"type": "text", "text": "The Kac-Peterson relation (2.1b) for $A_{r,k}$ takes the form ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "where $s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)$ is the Schur polynomial (see e.g. [27]) corresponding to the partition $(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))$ , and where $\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}$ for any weight $\\nu$ . In other words, $S_{\\lambda\\mu}/S_{0\\mu}$ is the Schur polynomial corresponding to $\\lambda$ , evaluated at roots of 1 determined by $\\mu$ . ", "page_idx": 8}, {"type": "text", "text": "The fusion (derived from the Pieri rule and (2.4)) ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "valid for $k\\geq2$ and $1\\leq\\ell<r$ , will be useful. ", "page_idx": 8}, {"type": "text", "text": "There are no exceptional fusion-symmetries for $A_{r}^{(1)}$ : ", "page_idx": 8}, {"type": "text", "text": "Theorem 3.A. The fusion-symmetries for $A_{r}^{(1)}$ level $k$ are $C^{i}\\pi[a]$ , for $i\\in\\{0,1\\}$ and any integer $0\\leq a\\leq r$ for which $1+k a$ is coprime to $r+1$ . ", "page_idx": 8}, {"type": "text", "text": "To avoid redundancies in the Theorem, for $r\\,=\\,1$ or $k\\,=\\,1$ take $i\\,=\\,0$ only. If we write ${\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}$ , where $r^{\\prime}$ is coprime to $k$ and $r^{\\prime\\prime}\|k^{\\infty}$ , then the number of simple-current automorphisms will exactly equal $r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})$ , where $\\varphi$ is the Euler totient. The $\\pi[a]$ commute with each other, and with $C$ . ", "page_idx": 8}, {"type": "text", "text": "For example, for $A_{1,k}$ when $k$ is odd, there is no nontrivial fusion-symmetry. When $k$ is even, there is exactly one, sending $\\lambda=\\lambda_{1}\\Lambda_{1}$ to $\\lambda$ (for $\\lambda_{1}$ even) or $J\\lambda=(k-\\lambda_{1})\\Lambda_{1}$ (for $\\lambda_{1}$ odd). For $A_{2,k}$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $k$ . 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The level ", "type": "text"}, {"bbox": [345, 234, 352, 243], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [353, 228, 459, 247], "score": 1.0, "content": " highest weights of ", "type": "text"}, {"bbox": [460, 228, 482, 244], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 228, 542, 247], "score": 1.0, "content": "constitute", "type": "text"}], "index": 7}, {"bbox": [70, 245, 541, 262], "spans": [{"bbox": [70, 245, 112, 262], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [113, 246, 129, 259], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [129, 245, 148, 262], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [148, 247, 155, 257], "score": 0.75, "content": "\\overline{r}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [156, 245, 195, 262], "score": 1.0, "content": "-tuples ", "type": "text"}, {"bbox": [195, 246, 284, 259], "score": 0.91, "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [284, 245, 466, 262], "score": 1.0, "content": " of non-negative integers obeying", "type": "text"}, {"bbox": [467, 245, 537, 261], "score": 0.91, "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "type": "inline_equation", "height": 16, "width": 70}, {"bbox": [537, 245, 541, 262], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [70, 259, 542, 276], "spans": [{"bbox": [70, 259, 392, 276], "score": 1.0, "content": "The Dynkin diagram symmetries form the dihedral group ", "type": "text"}, {"bbox": [392, 261, 451, 274], "score": 0.91, "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [451, 259, 542, 276], "score": 1.0, "content": "; it is generated", "type": "text"}], "index": 9}, {"bbox": [70, 274, 540, 290], "spans": [{"bbox": [70, 274, 213, 290], "score": 1.0, "content": "by the charge-conjugation ", "type": "text"}, {"bbox": [214, 275, 224, 285], "score": 0.83, "content": "C", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [225, 274, 332, 290], "score": 1.0, "content": " and simple-current ", "type": "text"}, {"bbox": [333, 276, 341, 285], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [341, 274, 395, 290], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [395, 274, 540, 288], "score": 0.89, "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "type": "inline_equation", "height": 14, "width": 145}], "index": 10}, {"bbox": [69, 290, 542, 309], "spans": [{"bbox": [69, 291, 94, 309], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 292, 235, 306], "score": 0.92, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 14, "width": 140}, {"bbox": [235, 291, 270, 309], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [270, 293, 366, 306], "score": 0.92, "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [367, 291, 388, 309], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [389, 290, 478, 308], "score": 0.92, "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "type": "inline_equation", "height": 18, "width": 89}, {"bbox": [479, 291, 542, 309], "score": 1.0, "content": ". 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In other words, ", "type": "text"}, {"bbox": [495, 414, 539, 426], "score": 0.94, "content": "S_{\\lambda\\mu}/S_{0\\mu}", "type": "inline_equation", "height": 12, "width": 44}], "index": 16}, {"bbox": [69, 425, 519, 443], "spans": [{"bbox": [69, 425, 289, 443], "score": 1.0, "content": "is the Schur polynomial corresponding to", "type": "text"}, {"bbox": [290, 428, 298, 438], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [298, 425, 505, 443], "score": 1.0, "content": ", evaluated at roots of 1 determined by ", "type": "text"}, {"bbox": [506, 432, 513, 440], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [514, 425, 519, 443], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [94, 439, 357, 453], "lines": [{"bbox": [96, 441, 355, 455], "spans": [{"bbox": [96, 441, 355, 455], "score": 1.0, "content": "The fusion (derived from the Pieri rule and (2.4))", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [71, 496, 304, 510], "lines": [{"bbox": [71, 498, 302, 511], "spans": [{"bbox": [71, 498, 118, 511], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 500, 147, 511], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [147, 498, 173, 511], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 499, 223, 511], "score": 0.9, "content": "1\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [224, 498, 302, 511], "score": 1.0, "content": ", will be useful.", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [95, 512, 371, 527], "lines": [{"bbox": [93, 510, 372, 529], "spans": [{"bbox": [93, 512, 344, 529], "score": 1.0, "content": "There are no exceptional fusion-symmetries for ", "type": "text"}, {"bbox": [345, 510, 367, 527], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [367, 512, 372, 529], "score": 1.0, "content": ":", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [70, 533, 542, 565], "lines": [{"bbox": [93, 533, 543, 554], "spans": [{"bbox": [93, 534, 323, 554], "score": 1.0, "content": "Theorem 3.A. The fusion-symmetries for ", "type": "text"}, {"bbox": [323, 533, 346, 550], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [347, 534, 375, 554], "score": 1.0, "content": "level", "type": "text"}, {"bbox": [375, 537, 384, 549], "score": 0.72, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [384, 534, 407, 554], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [407, 536, 442, 551], "score": 0.92, "content": "C^{i}\\pi[a]", "type": "inline_equation", "height": 15, "width": 35}, {"bbox": [442, 534, 467, 554], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [467, 537, 517, 551], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [517, 534, 543, 554], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [72, 551, 383, 568], "spans": [{"bbox": [72, 551, 133, 568], "score": 1.0, "content": "any integer ", "type": "text"}, {"bbox": [134, 553, 184, 565], "score": 0.86, "content": "0\\leq a\\leq r", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [185, 551, 240, 568], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [240, 552, 275, 564], "score": 0.88, "content": "1+k a", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [275, 551, 350, 568], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [350, 552, 378, 564], "score": 0.9, "content": "r+1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [378, 551, 383, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [70, 571, 541, 628], "lines": [{"bbox": [94, 573, 541, 587], "spans": [{"bbox": [94, 574, 329, 587], "score": 1.0, "content": "To avoid redundancies in the Theorem, for ", "type": "text"}, {"bbox": [330, 574, 362, 584], "score": 0.89, "content": "r\\,=\\,1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [362, 574, 381, 587], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [381, 573, 414, 585], "score": 0.89, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [415, 574, 444, 587], "score": 1.0, "content": " take ", "type": "text"}, {"bbox": [445, 574, 475, 585], "score": 0.88, "content": "i\\,=\\,0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [476, 574, 541, 587], "score": 1.0, "content": " only. If we", "type": "text"}], "index": 24}, {"bbox": [72, 588, 541, 601], "spans": [{"bbox": [72, 588, 102, 601], "score": 1.0, "content": "write ", "type": "text"}, {"bbox": [102, 589, 149, 599], "score": 0.91, "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [149, 588, 191, 601], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [192, 589, 201, 598], "score": 0.9, "content": "r^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [201, 588, 280, 601], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [280, 590, 287, 599], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [288, 588, 315, 601], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [316, 588, 348, 601], "score": 0.92, "content": "r^{\\prime\\prime}\|k^{\\infty}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [348, 588, 541, 601], "score": 1.0, "content": ", then the number of simple-current", "type": "text"}], "index": 25}, {"bbox": [70, 601, 542, 617], "spans": [{"bbox": [70, 602, 246, 617], "score": 1.0, "content": "automorphisms will exactly equal ", "type": "text"}, {"bbox": [246, 603, 289, 616], "score": 0.93, "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [290, 602, 329, 617], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [329, 605, 338, 615], "score": 0.81, "content": "\\varphi", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [338, 602, 468, 617], "score": 1.0, "content": " is the Euler totient. The ", "type": "text"}, {"bbox": [468, 601, 489, 616], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [489, 602, 542, 617], "score": 1.0, "content": " commute", "type": "text"}], "index": 26}, {"bbox": [72, 617, 224, 630], "spans": [{"bbox": [72, 617, 210, 630], "score": 1.0, "content": "with each other, and with ", "type": "text"}, {"bbox": [210, 618, 220, 627], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [220, 617, 224, 630], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [70, 630, 541, 686], "lines": [{"bbox": [95, 631, 540, 646], "spans": [{"bbox": [95, 631, 184, 646], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [185, 633, 207, 645], "score": 0.93, "content": "A_{1,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [207, 631, 241, 646], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [242, 633, 248, 641], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [249, 631, 532, 646], "score": 1.0, "content": " is odd, there is no nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [532, 633, 540, 642], "score": 0.83, "content": "k", "type": "inline_equation", "height": 9, "width": 8}], "index": 28}, {"bbox": [69, 645, 541, 661], "spans": [{"bbox": [69, 645, 265, 661], "score": 1.0, "content": "is even, there is exactly one, sending ", "type": "text"}, {"bbox": [266, 647, 315, 658], "score": 0.93, "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [315, 645, 333, 661], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [333, 647, 340, 656], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [340, 645, 366, 661], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [367, 647, 379, 658], "score": 0.9, "content": "\\lambda_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [379, 645, 429, 661], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [429, 646, 516, 659], "score": 0.93, "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [517, 645, 541, 661], "score": 1.0, "content": " (for", "type": "text"}], "index": 29}, {"bbox": [71, 659, 541, 675], "spans": [{"bbox": [71, 662, 83, 672], "score": 0.89, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [84, 659, 144, 675], "score": 1.0, "content": " odd). For ", "type": "text"}, {"bbox": [145, 662, 167, 674], "score": 0.93, "content": "A_{2,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [167, 659, 541, 675], "score": 1.0, "content": ", there are either six or four fusion-symmetries, depending on whether", "type": "text"}], "index": 30}, {"bbox": [70, 674, 169, 689], "spans": [{"bbox": [70, 674, 157, 689], "score": 1.0, "content": "or not 3 divides ", "type": "text"}, {"bbox": [158, 676, 164, 685], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [165, 674, 169, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [87, 352, 487, 383], "lines": [{"bbox": [87, 352, 487, 383], "spans": [{"bbox": [87, 352, 487, 383], "score": 0.85, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19}], "discarded_blocks": [{"type": "discarded", "bbox": [200, 71, 410, 85], "lines": [{"bbox": [201, 73, 410, 86], "spans": [{"bbox": [201, 73, 410, 86], "score": 1.0, "content": "3. Data for the Affine Algebras.", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 99, 542, 191], "lines": [{"bbox": [95, 102, 541, 117], "spans": [{"bbox": [95, 102, 541, 117], "score": 1.0, "content": "Our main task in this paper is to find and construct all fusion-symmetries for the", "type": "text"}], "index": 0}, {"bbox": [69, 113, 544, 135], "spans": [{"bbox": [69, 113, 151, 135], "score": 1.0, "content": "affine algebras ", "type": "text"}, {"bbox": [151, 116, 174, 130], "score": 0.94, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [175, 113, 239, 135], "score": 1.0, "content": ", for simple ", "type": "text"}, {"bbox": [239, 120, 255, 131], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [255, 113, 544, 135], "score": 1.0, "content": ". In this section we state the results, and in the next", "type": "text"}], "index": 1}, {"bbox": [70, 132, 542, 148], "spans": [{"bbox": [70, 132, 542, 148], "score": 1.0, "content": "section we prove the completeness of our lists. Recall the simple-current automorphism", "type": "text"}], "index": 2}, {"bbox": [71, 146, 541, 162], "spans": [{"bbox": [71, 148, 92, 161], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [92, 146, 231, 162], "score": 1.0, "content": " and Galois automorphism ", "type": "text"}, {"bbox": [231, 148, 256, 161], "score": 0.91, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [257, 146, 429, 162], "score": 1.0, "content": " defined in \u00a72.3, and the notation ", "type": "text"}, {"bbox": [429, 148, 484, 159], "score": 0.94, "content": "\\kappa=k\\!+\\!h^{\\vee}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [485, 146, 541, 162], "score": 1.0, "content": ". It will be", "type": "text"}], "index": 3}, {"bbox": [68, 161, 541, 178], "spans": [{"bbox": [68, 163, 178, 178], "score": 1.0, "content": "convenient to write ", "type": "text"}, {"bbox": [179, 164, 206, 178], "score": 0.86, "content": "X{_{r,k}}^{,}", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [206, 163, 230, 178], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [231, 161, 255, 176], "score": 0.9, "content": "\\cdot X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [257, 163, 310, 178], "score": 1.0, "content": "and level ", "type": "text"}, {"bbox": [310, 165, 320, 174], "score": 0.62, "content": "k'", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [321, 163, 380, 178], "score": 1.0, "content": ". We write ", "type": "text"}, {"bbox": [381, 166, 389, 174], "score": 0.91, "content": "_S", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [389, 163, 541, 178], "score": 1.0, "content": " for the group of symmetries", "type": "text"}], "index": 4}, {"bbox": [70, 177, 245, 192], "spans": [{"bbox": [70, 177, 245, 192], "score": 1.0, "content": "of the extended Dynkin diagram.", "type": "text"}], "index": 5}], "index": 2.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [68, 102, 544, 192]}, {"type": "title", "bbox": [71, 203, 218, 221], "lines": [{"bbox": [69, 202, 218, 225], "spans": [{"bbox": [69, 202, 160, 225], "score": 1.0, "content": "3.1. The algebra ", "type": "text"}, {"bbox": [161, 205, 183, 221], "score": 0.48, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [183, 202, 189, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 208, 218, 221], "score": 0.48, "content": "r\\geq1", "type": "inline_equation", "height": 13, "width": 28}], "index": 6}], "index": 6, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 228, 541, 323], "lines": [{"bbox": [93, 228, 542, 247], "spans": [{"bbox": [93, 228, 133, 247], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [133, 232, 190, 243], "score": 0.89, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [190, 228, 219, 247], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [219, 233, 277, 243], "score": 0.92, "content": "\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}", "type": "inline_equation", "height": 10, "width": 58}, {"bbox": [278, 228, 345, 247], "score": 1.0, "content": ". The level ", "type": "text"}, {"bbox": [345, 234, 352, 243], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [353, 228, 459, 247], "score": 1.0, "content": " highest weights of ", "type": "text"}, {"bbox": [460, 228, 482, 244], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 228, 542, 247], "score": 1.0, "content": "constitute", "type": "text"}], "index": 7}, {"bbox": [70, 245, 541, 262], "spans": [{"bbox": [70, 245, 112, 262], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [113, 246, 129, 259], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [129, 245, 148, 262], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [148, 247, 155, 257], "score": 0.75, "content": "\\overline{r}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [156, 245, 195, 262], "score": 1.0, "content": "-tuples ", "type": "text"}, {"bbox": [195, 246, 284, 259], "score": 0.91, "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [284, 245, 466, 262], "score": 1.0, "content": " of non-negative integers obeying", "type": "text"}, {"bbox": [467, 245, 537, 261], "score": 0.91, "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "type": "inline_equation", "height": 16, "width": 70}, {"bbox": [537, 245, 541, 262], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [70, 259, 542, 276], "spans": [{"bbox": [70, 259, 392, 276], "score": 1.0, "content": "The Dynkin diagram symmetries form the dihedral group ", "type": "text"}, {"bbox": [392, 261, 451, 274], "score": 0.91, "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [451, 259, 542, 276], "score": 1.0, "content": "; it is generated", "type": "text"}], "index": 9}, {"bbox": [70, 274, 540, 290], "spans": [{"bbox": [70, 274, 213, 290], "score": 1.0, "content": "by the charge-conjugation ", "type": "text"}, {"bbox": [214, 275, 224, 285], "score": 0.83, "content": "C", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [225, 274, 332, 290], "score": 1.0, "content": " and simple-current ", "type": "text"}, {"bbox": [333, 276, 341, 285], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [341, 274, 395, 290], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [395, 274, 540, 288], "score": 0.89, "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "type": "inline_equation", "height": 14, "width": 145}], "index": 10}, {"bbox": [69, 290, 542, 309], "spans": [{"bbox": [69, 291, 94, 309], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 292, 235, 306], "score": 0.92, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 14, "width": 140}, {"bbox": [235, 291, 270, 309], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [270, 293, 366, 306], "score": 0.92, "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [367, 291, 388, 309], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [389, 290, 478, 308], "score": 0.92, "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "type": "inline_equation", "height": 18, "width": 89}, {"bbox": [479, 291, 542, 309], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 11}, {"bbox": [71, 309, 159, 325], "spans": [{"bbox": [71, 313, 108, 322], "score": 0.88, "content": "C=i d", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [108, 310, 132, 325], "score": 1.0, "content": ". for ", "type": "text"}, {"bbox": [132, 309, 154, 325], "score": 0.93, "content": "{A}_{1}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [155, 310, 159, 325], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [69, 228, 542, 325]}, {"type": "text", "bbox": [95, 324, 394, 339], "lines": [{"bbox": [94, 325, 393, 341], "spans": [{"bbox": [94, 325, 290, 341], "score": 1.0, "content": "The Kac-Peterson relation (2.1b) for ", "type": "text"}, {"bbox": [290, 328, 313, 340], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [313, 325, 393, 341], "score": 1.0, "content": " takes the form", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [94, 325, 393, 341]}, {"type": "interline_equation", "bbox": [87, 352, 487, 383], "lines": [{"bbox": [87, 352, 487, 383], "spans": [{"bbox": [87, 352, 487, 383], "score": 0.85, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 395, 540, 439], "lines": [{"bbox": [71, 397, 540, 414], "spans": [{"bbox": [71, 397, 105, 414], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 399, 197, 412], "score": 0.92, "content": "s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [198, 397, 540, 414], "score": 1.0, "content": " is the Schur polynomial (see e.g. [27]) corresponding to the parti-", "type": "text"}], "index": 15}, {"bbox": [69, 411, 539, 430], "spans": [{"bbox": [69, 411, 96, 430], "score": 1.0, "content": "tion ", "type": "text"}, {"bbox": [96, 413, 176, 426], "score": 0.9, "content": "(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [176, 411, 239, 430], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [239, 411, 317, 427], "score": 0.94, "content": "\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}", "type": "inline_equation", "height": 16, "width": 78}, {"bbox": [317, 411, 397, 430], "score": 1.0, "content": " for any weight ", "type": "text"}, {"bbox": [397, 415, 405, 423], "score": 0.72, "content": "\\nu", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [405, 411, 495, 430], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [495, 414, 539, 426], "score": 0.94, "content": "S_{\\lambda\\mu}/S_{0\\mu}", "type": "inline_equation", "height": 12, "width": 44}], "index": 16}, {"bbox": [69, 425, 519, 443], "spans": [{"bbox": [69, 425, 289, 443], "score": 1.0, "content": "is the Schur polynomial corresponding to", "type": "text"}, {"bbox": [290, 428, 298, 438], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [298, 425, 505, 443], "score": 1.0, "content": ", evaluated at roots of 1 determined by ", "type": "text"}, {"bbox": [506, 432, 513, 440], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [514, 425, 519, 443], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [69, 397, 540, 443]}, {"type": "text", "bbox": [94, 439, 357, 453], "lines": [{"bbox": [96, 441, 355, 455], "spans": [{"bbox": [96, 441, 355, 455], "score": 1.0, "content": "The fusion (derived from the Pieri rule and (2.4))", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [96, 441, 355, 455]}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 496, 304, 510], "lines": [{"bbox": [71, 498, 302, 511], "spans": [{"bbox": [71, 498, 118, 511], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 500, 147, 511], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [147, 498, 173, 511], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 499, 223, 511], "score": 0.9, "content": "1\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [224, 498, 302, 511], "score": 1.0, "content": ", will be useful.", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [71, 498, 302, 511]}, {"type": "text", "bbox": [95, 512, 371, 527], "lines": [{"bbox": [93, 510, 372, 529], "spans": [{"bbox": [93, 512, 344, 529], "score": 1.0, "content": "There are no exceptional fusion-symmetries for ", "type": "text"}, {"bbox": [345, 510, 367, 527], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [367, 512, 372, 529], "score": 1.0, "content": ":", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [93, 510, 372, 529]}, {"type": "text", "bbox": [70, 533, 542, 565], "lines": [{"bbox": [93, 533, 543, 554], "spans": [{"bbox": [93, 534, 323, 554], "score": 1.0, "content": "Theorem 3.A. The fusion-symmetries for ", "type": "text"}, {"bbox": [323, 533, 346, 550], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [347, 534, 375, 554], "score": 1.0, "content": "level", "type": "text"}, {"bbox": [375, 537, 384, 549], "score": 0.72, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [384, 534, 407, 554], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [407, 536, 442, 551], "score": 0.92, "content": "C^{i}\\pi[a]", "type": "inline_equation", "height": 15, "width": 35}, {"bbox": [442, 534, 467, 554], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [467, 537, 517, 551], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [517, 534, 543, 554], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [72, 551, 383, 568], "spans": [{"bbox": [72, 551, 133, 568], "score": 1.0, "content": "any integer ", "type": "text"}, {"bbox": [134, 553, 184, 565], "score": 0.86, "content": "0\\leq a\\leq r", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [185, 551, 240, 568], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [240, 552, 275, 564], "score": 0.88, "content": "1+k a", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [275, 551, 350, 568], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [350, 552, 378, 564], "score": 0.9, "content": "r+1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [378, 551, 383, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [72, 533, 543, 568]}, {"type": "text", "bbox": [70, 571, 541, 628], "lines": [{"bbox": [94, 573, 541, 587], "spans": [{"bbox": [94, 574, 329, 587], "score": 1.0, "content": "To avoid redundancies in the Theorem, for ", "type": "text"}, {"bbox": [330, 574, 362, 584], "score": 0.89, "content": "r\\,=\\,1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [362, 574, 381, 587], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [381, 573, 414, 585], "score": 0.89, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [415, 574, 444, 587], "score": 1.0, "content": " take ", "type": "text"}, {"bbox": [445, 574, 475, 585], "score": 0.88, "content": "i\\,=\\,0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [476, 574, 541, 587], "score": 1.0, "content": " only. If we", "type": "text"}], "index": 24}, {"bbox": [72, 588, 541, 601], "spans": [{"bbox": [72, 588, 102, 601], "score": 1.0, "content": "write ", "type": "text"}, {"bbox": [102, 589, 149, 599], "score": 0.91, "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [149, 588, 191, 601], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [192, 589, 201, 598], "score": 0.9, "content": "r^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [201, 588, 280, 601], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [280, 590, 287, 599], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [288, 588, 315, 601], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [316, 588, 348, 601], "score": 0.92, "content": "r^{\\prime\\prime}\|k^{\\infty}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [348, 588, 541, 601], "score": 1.0, "content": ", then the number of simple-current", "type": "text"}], "index": 25}, {"bbox": [70, 601, 542, 617], "spans": [{"bbox": [70, 602, 246, 617], "score": 1.0, "content": "automorphisms will exactly equal ", "type": "text"}, {"bbox": [246, 603, 289, 616], "score": 0.93, "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [290, 602, 329, 617], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [329, 605, 338, 615], "score": 0.81, "content": "\\varphi", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [338, 602, 468, 617], "score": 1.0, "content": " is the Euler totient. The ", "type": "text"}, {"bbox": [468, 601, 489, 616], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [489, 602, 542, 617], "score": 1.0, "content": " commute", "type": "text"}], "index": 26}, {"bbox": [72, 617, 224, 630], "spans": [{"bbox": [72, 617, 210, 630], "score": 1.0, "content": "with each other, and with ", "type": "text"}, {"bbox": [210, 618, 220, 627], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [220, 617, 224, 630], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 573, 542, 630]}, {"type": "text", "bbox": [70, 630, 541, 686], "lines": [{"bbox": [95, 631, 540, 646], "spans": [{"bbox": [95, 631, 184, 646], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [185, 633, 207, 645], "score": 0.93, "content": "A_{1,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [207, 631, 241, 646], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [242, 633, 248, 641], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [249, 631, 532, 646], "score": 1.0, "content": " is odd, there is no nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [532, 633, 540, 642], "score": 0.83, "content": "k", "type": "inline_equation", "height": 9, "width": 8}], "index": 28}, {"bbox": [69, 645, 541, 661], "spans": [{"bbox": [69, 645, 265, 661], "score": 1.0, "content": "is even, there is exactly one, sending ", "type": "text"}, {"bbox": [266, 647, 315, 658], "score": 0.93, "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [315, 645, 333, 661], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [333, 647, 340, 656], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [340, 645, 366, 661], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [367, 647, 379, 658], "score": 0.9, "content": "\\lambda_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [379, 645, 429, 661], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [429, 646, 516, 659], "score": 0.93, "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [517, 645, 541, 661], "score": 1.0, "content": " (for", "type": "text"}], "index": 29}, {"bbox": [71, 659, 541, 675], "spans": [{"bbox": [71, 662, 83, 672], "score": 0.89, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [84, 659, 144, 675], "score": 1.0, "content": " odd). For ", "type": "text"}, {"bbox": [145, 662, 167, 674], "score": 0.93, "content": "A_{2,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [167, 659, 541, 675], "score": 1.0, "content": ", there are either six or four fusion-symmetries, depending on whether", "type": "text"}], "index": 30}, {"bbox": [70, 674, 169, 689], "spans": [{"bbox": [70, 674, 157, 689], "score": 1.0, "content": "or not 3 divides ", "type": "text"}, {"bbox": [158, 676, 164, 685], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [165, 674, 169, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [69, 631, 541, 689]}]}
0002044v1	9	3.2. The algebra $$B_{r}^{(1)}$$ , $$r\geq3$$ A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+\lambda_{1}+2\lambda_{2}+\cdot\cdot\cdot+2\lambda_{r-1}+\lambda_{r}$$ , and $$\kappa=k+2r-1$$ . The charge-conjugation is trivial, but there is a simple-current: $$J\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},.\ldots,\lambda_{r}\right)$$ . It has $$Q(\lambda)=\lambda_{r}/2$$ . The only fusion products we need are for all $$1\leq i<r-1$$ , $$k>2$$ , and $$0<\ell<k$$ , where we drop $$\mathrm{\Delta}^{\prime}\Lambda_{r-1}+(\ell-2)\Lambda_{r}{}^{\prime}$$ if $$\ell=1$$ . We will also use the character formula (2.1b) where $$\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$$ and For $$k\,=\,2$$ ( $$\kappa\,=\,2r+1$$ ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define $$\gamma^{i}=\gamma^{\kappa-i}=\Lambda_{i}$$ for $$i\,=\,1,2,\dots,r\,-\,1$$ , and $$\gamma^{r}\,=\,\gamma^{r+1}\,=\,2\Lambda_{r}$$ . Then for any $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ coprime to $$\kappa$$ , $$\pi\{m\}$$ fixes $$0$$ and $$J$$ , sends $$\gamma^{a}$$ to $$\gamma^{m a}$$ (where the superscript is taken mod $$\kappa$$ ), and stabilises $$\{\Lambda_{r},J\Lambda_{r}\}$$ $$(\pi\{m\}\Lambda_{r}=\Lambda_{r}$$ iff the Jacobi symbol $$\Bigl(\frac{\kappa}{m}\Bigr)$$ equals $$+1$$ ). Why is $$k=2$$ so special here? One reason is that rank-level duality associates $$B_{r,2}$$ with $$\mathrm{u}(1)_{2r+1}$$ , and it is easy to confirm that $$\widehat{\mathrm{u(1)}}$$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $$B_{r,2}$$ matrix $$S$$ formally looks like the character table of the dihedral group and for some $$r$$ actually equals the Kac-Peterson matrix $$S$$ associated to the dihedral group $${\mathfrak{D}}_{{\sqrt{\kappa}}}$$ twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data. Theorem 3.B. The fusion-symmetries of $$B_{r}^{(1)}$$ level $$k$$ for $$k\ \neq\ 2$$ are $$\pi[1]^{i}$$ where $$i\in\{0,1\}$$ . For $$k=2$$ a fusion-symmetry will equal $$\pi[1]^{i}\,\pi\{m\}$$ for $$i\in\{0,1\}$$ and $$m\in\mathbb{Z}_{\kappa}^{\times}$$ , $$1\leq m\leq r$$ . When $$k=1$$ , $$\pi[1]$$ is trivial. We have $$\mathcal{F}(B_{r,2})\cong\mathbb{Z}_{2}\times(\mathbb{Z}_{2r+1}^{\times}/\{\pm1\}).$$ 3.3. The algebra $$C_{r}^{(1)}$$ , $$r\geq2$$ A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}+\lambda_{1}+\cdot\cdot\cdot+\lambda_{r}$$ and $$\kappa=k+r+1$$ . Charge-conjugation $$C$$ again is trivial, and there is a simple-current $$J$$ defined by $$J\lambda=\left(\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1},\lambda_{0}\right)$$ , with $$\begin{array}{r}{Q(\lambda)=(\sum_{j=1}^{r}j\lambda_{j})/2}\end{array}$$ . Choose any $$\lambda\:\in\:P_{+}$$ . The Young diagram for $$\lambda$$ is defined in the usual way: for $$1\leq\ell\leq r$$ , the $$\ell$$ th row consists of $$\begin{array}{r}{\lambda(\ell)\overset{\mathrm{def}}{=}\sum_{i=\ell}^{r}\lambda_{i}}\end{array}$$ boxes. Let $$\tau\lambda$$ denote the $$C_{k,r}$$ weight	<p>3.2. The algebra $$B_{r}^{(1)}$$ , $$r\geq3$$</p> <p>A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+\lambda_{1}+2\lambda_{2}+\cdot\cdot\cdot+2\lambda_{r-1}+\lambda_{r}$$ , and $$\kappa=k+2r-1$$ . The charge-conjugation is trivial, but there is a simple-current: $$J\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},.\ldots,\lambda_{r}\right)$$ . It has $$Q(\lambda)=\lambda_{r}/2$$ .</p> <p>The only fusion products we need are</p> <p>for all $$1\leq i<r-1$$ , $$k>2$$ , and $$0<\ell<k$$ , where we drop $$\mathrm{\Delta}^{\prime}\Lambda_{r-1}+(\ell-2)\Lambda_{r}{}^{\prime}$$ if $$\ell=1$$ . We will also use the character formula (2.1b)</p> <p>where $$\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$$ and</p> <p>For $$k\,=\,2$$ ( $$\kappa\,=\,2r+1$$ ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define $$\gamma^{i}=\gamma^{\kappa-i}=\Lambda_{i}$$ for $$i\,=\,1,2,\dots,r\,-\,1$$ , and $$\gamma^{r}\,=\,\gamma^{r+1}\,=\,2\Lambda_{r}$$ . Then for any $${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$$ coprime to $$\kappa$$ , $$\pi\{m\}$$ fixes $$0$$ and $$J$$ , sends $$\gamma^{a}$$ to $$\gamma^{m a}$$ (where the superscript is taken mod $$\kappa$$ ), and stabilises $$\{\Lambda_{r},J\Lambda_{r}\}$$ $$(\pi\{m\}\Lambda_{r}=\Lambda_{r}$$ iff the Jacobi symbol $$\Bigl(\frac{\kappa}{m}\Bigr)$$ equals $$+1$$ ).</p> <p>Why is $$k=2$$ so special here? One reason is that rank-level duality associates $$B_{r,2}$$ with $$\mathrm{u}(1)_{2r+1}$$ , and it is easy to confirm that $$\widehat{\mathrm{u(1)}}$$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $$B_{r,2}$$ matrix $$S$$ formally looks like the character table of the dihedral group and for some $$r$$ actually equals the Kac-Peterson matrix $$S$$ associated to the dihedral group $${\mathfrak{D}}_{{\sqrt{\kappa}}}$$ twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data.</p> <p>Theorem 3.B. The fusion-symmetries of $$B_{r}^{(1)}$$ level $$k$$ for $$k\ \neq\ 2$$ are $$\pi[1]^{i}$$ where $$i\in\{0,1\}$$ . For $$k=2$$ a fusion-symmetry will equal $$\pi[1]^{i}\,\pi\{m\}$$ for $$i\in\{0,1\}$$ and $$m\in\mathbb{Z}_{\kappa}^{\times}$$ , $$1\leq m\leq r$$ .</p> <p>When $$k=1$$ , $$\pi[1]$$ is trivial. We have $$\mathcal{F}(B_{r,2})\cong\mathbb{Z}_{2}\times(\mathbb{Z}_{2r+1}^{\times}/\{\pm1\}).$$</p> <p>3.3. The algebra $$C_{r}^{(1)}$$ , $$r\geq2$$</p> <p>A weight $$\lambda$$ of $$P_{+}$$ satisfies $$k=\lambda_{0}+\lambda_{1}+\cdot\cdot\cdot+\lambda_{r}$$ and $$\kappa=k+r+1$$ . Charge-conjugation $$C$$ again is trivial, and there is a simple-current $$J$$ defined by $$J\lambda=\left(\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1},\lambda_{0}\right)$$ , with $$\begin{array}{r}{Q(\lambda)=(\sum_{j=1}^{r}j\lambda_{j})/2}\end{array}$$ .</p> <p>Choose any $$\lambda\:\in\:P_{+}$$ . The Young diagram for $$\lambda$$ is defined in the usual way: for $$1\leq\ell\leq r$$ , the $$\ell$$ th row consists of $$\begin{array}{r}{\lambda(\ell)\overset{\mathrm{def}}{=}\sum_{i=\ell}^{r}\lambda_{i}}\end{array}$$ boxes. Let $$\tau\lambda$$ denote the $$C_{k,r}$$ weight</p>	[{"type": "text", "coordinates": [70, 69, 221, 86], "content": "3.2. The algebra $$B_{r}^{(1)}$$ , $$r\\geq3$$", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 92, 542, 136], "content": "A weight $$\\lambda$$ in $$P_{+}$$ satisfies $$k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}$$ , and $$\\kappa=k+2r-1$$ .\nThe charge-conjugation is trivial, but there is a simple-current: $$J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)$$ .\nIt has $$Q(\\lambda)=\\lambda_{r}/2$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [94, 137, 294, 151], "content": "The only fusion products we need are", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [69, 205, 541, 235], "content": "for all $$1\\leq i<r-1$$ , $$k>2$$ , and $$0<\\ell<k$$ , where we drop $$\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}$$ if $$\\ell=1$$ . We\nwill also use the character formula (2.1b)", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [195, 246, 415, 285], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 295, 227, 310], "content": "where $$\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$$ and", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [249, 309, 362, 349], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 353, 541, 426], "content": "For $$k\\,=\\,2$$ ( $$\\kappa\\,=\\,2r+1$$ ) there are several Galois fusion-symmetries \u2014 one for each\nGalois automorphism, since S020 = 41\u03ba is rational. In particular, define $$\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}$$ for\n$$i\\,=\\,1,2,\\dots,r\\,-\\,1$$ , and $$\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}$$ . Then for any $${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$$ coprime to $$\\kappa$$ , $$\\pi\\{m\\}$$ fixes $$0$$\nand $$J$$ , sends $$\\gamma^{a}$$ to $$\\gamma^{m a}$$ (where the superscript is taken mod $$\\kappa$$ ), and stabilises $$\\{\\Lambda_{r},J\\Lambda_{r}\\}$$\n$$(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}$$ iff the Jacobi symbol $$\\Bigl(\\frac{\\kappa}{m}\\Bigr)$$ equals $$+1$$ ).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 426, 541, 530], "content": "Why is $$k=2$$ so special here? One reason is that rank-level duality associates $$B_{r,2}$$ with\n$$\\mathrm{u}(1)_{2r+1}$$ , and it is easy to confirm that $$\\widehat{\\mathrm{u(1)}}$$ has a rich variety of fusion-symmetries (and\nmodular invariants) coming from its si mple-currents. Also, the $$B_{r,2}$$ matrix $$S$$ formally\nlooks like the character table of the dihedral group and for some $$r$$ actually equals the\nKac-Peterson matrix $$S$$ associated to the dihedral group $${\\mathfrak{D}}_{{\\sqrt{\\kappa}}}$$ twisted by an appropriate 3-\ncocycle [5] \u2014 finite group modular data tends to have significantly more modular invariants\nand fusion-symmetries than e.g. affine modular data.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 535, 542, 581], "content": "Theorem 3.B. The fusion-symmetries of $$B_{r}^{(1)}$$ level $$k$$ for $$k\\ \\neq\\ 2$$ are $$\\pi[1]^{i}$$ where\n$$i\\in\\{0,1\\}$$ . For $$k=2$$ a fusion-symmetry will equal $$\\pi[1]^{i}\\,\\pi\\{m\\}$$ for $$i\\in\\{0,1\\}$$ and $$m\\in\\mathbb{Z}_{\\kappa}^{\\times}$$ ,\n$$1\\leq m\\leq r$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [93, 586, 451, 604], "content": "When $$k=1$$ , $$\\pi[1]$$ is trivial. We have $$\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).$$", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [71, 616, 220, 633], "content": "3.3. The algebra $$C_{r}^{(1)}$$ , $$r\\geq2$$", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [71, 639, 541, 683], "content": "A weight $$\\lambda$$ of $$P_{+}$$ satisfies $$k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}$$ and $$\\kappa=k+r+1$$ . Charge-conjugation\n$$C$$ again is trivial, and there is a simple-current $$J$$ defined by $$J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)$$ ,\nwith $$\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [70, 683, 540, 716], "content": "Choose any $$\\lambda\\:\\in\\:P_{+}$$ . The Young diagram for $$\\lambda$$ is defined in the usual way: for\n$$1\\leq\\ell\\leq r$$ , the $$\\ell$$ th row consists of $$\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}$$ boxes. Let $$\\tau\\lambda$$ denote the $$C_{k,r}$$ weight", "block_type": "text", "index": 14}]	[{"type": "text", "coordinates": [68, 67, 160, 92], "content": "3.2. The algebra ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [161, 72, 183, 86], "content": "B_{r}^{(1)}", "score": 0.73, "index": 2}, {"type": "text", "coordinates": [184, 67, 190, 92], "content": ", ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [190, 77, 219, 87], "content": "r\\geq3", "score": 0.74, "index": 4}, {"type": "text", "coordinates": [95, 95, 145, 110], "content": "A weight ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [145, 96, 154, 106], "content": "\\lambda", "score": 0.8, "index": 6}, {"type": "text", "coordinates": [154, 95, 170, 110], "content": " in ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [171, 96, 187, 109], "content": "P_{+}", "score": 0.9, "index": 8}, {"type": "text", "coordinates": [187, 95, 233, 110], "content": " satisfies", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [234, 95, 429, 108], "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "score": 0.91, "index": 10}, {"type": "text", "coordinates": [430, 95, 459, 110], "content": ", and ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [459, 96, 537, 107], "content": "\\kappa=k+2r-1", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [537, 95, 541, 110], "content": ".", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [70, 109, 408, 125], "content": "The charge-conjugation is trivial, but there is a simple-current: ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [408, 109, 536, 123], "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "score": 0.91, "index": 15}, {"type": "text", "coordinates": [536, 109, 541, 125], "content": ".", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [69, 123, 105, 140], "content": "It has ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [105, 123, 172, 138], "content": "Q(\\lambda)=\\lambda_{r}/2", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [172, 123, 176, 140], "content": ".", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [95, 138, 294, 152], "content": "The only fusion products we need are", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [70, 208, 105, 223], "content": "for all ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [106, 209, 175, 221], "content": "1\\leq i<r-1", "score": 0.83, "index": 22}, {"type": "text", "coordinates": [175, 208, 180, 223], "content": ",", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [181, 208, 210, 221], "content": "k>2", "score": 0.68, "index": 24}, {"type": "text", "coordinates": [211, 208, 240, 223], "content": ", and ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [240, 208, 290, 221], "content": "0<\\ell<k", "score": 0.91, "index": 26}, {"type": "text", "coordinates": [291, 208, 378, 223], "content": ", where we drop ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [379, 208, 473, 222], "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "score": 0.91, "index": 28}, {"type": "text", "coordinates": [473, 208, 487, 223], "content": " if ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [487, 209, 514, 219], "content": "\\ell=1", "score": 0.87, "index": 30}, {"type": "text", "coordinates": [515, 208, 541, 223], "content": ". We", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [71, 223, 288, 238], "content": "will also use the character formula (2.1b)", "score": 1.0, "index": 32}, {"type": "interline_equation", "coordinates": [195, 246, 415, 285], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "score": 0.94, "index": 33}, {"type": "text", "coordinates": [72, 297, 105, 312], "content": "where ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [106, 298, 203, 311], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "score": 0.94, "index": 35}, {"type": "text", "coordinates": [203, 297, 227, 312], "content": " and", "score": 1.0, "index": 36}, {"type": "interline_equation", "coordinates": [249, 309, 362, 349], "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "score": 0.94, "index": 37}, {"type": "text", "coordinates": [93, 354, 117, 371], "content": "For ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [117, 358, 150, 367], "content": "k\\,=\\,2", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [150, 354, 158, 371], "content": " (", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [158, 357, 219, 368], "content": "\\kappa\\,=\\,2r+1", "score": 0.69, "index": 41}, {"type": "text", "coordinates": [220, 354, 541, 371], "content": ") there are several Galois fusion-symmetries \u2014 one for each", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [69, 368, 272, 388], "content": "Galois automorphism, since S020 = 41\u03ba ", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [267, 369, 441, 387], "content": "is rational. In particular, define ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [441, 371, 521, 384], "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "score": 0.94, "index": 45}, {"type": "text", "coordinates": [521, 369, 541, 387], "content": " for", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [71, 387, 166, 398], "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "score": 0.9, "index": 47}, {"type": "text", "coordinates": [166, 383, 198, 402], "content": ", and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [198, 385, 293, 398], "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [293, 383, 378, 402], "content": ". Then for any ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [379, 390, 389, 396], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "score": 0.85, "index": 51}, {"type": "text", "coordinates": [390, 383, 455, 402], "content": " coprime to ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [456, 390, 463, 396], "content": "\\kappa", "score": 0.69, "index": 53}, {"type": "text", "coordinates": [463, 383, 470, 402], "content": ", ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [471, 386, 501, 398], "content": "\\pi\\{m\\}", "score": 0.93, "index": 55}, {"type": "text", "coordinates": [501, 383, 533, 402], "content": " fixes ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [533, 387, 540, 396], "content": "0", "score": 0.39, "index": 57}, {"type": "text", "coordinates": [71, 399, 95, 414], "content": "and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [95, 401, 103, 410], "content": "J", "score": 0.89, "index": 59}, {"type": "text", "coordinates": [103, 399, 142, 414], "content": ", sends ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [142, 401, 155, 412], "content": "\\gamma^{a}", "score": 0.91, "index": 61}, {"type": "text", "coordinates": [155, 399, 173, 414], "content": " to ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [174, 401, 195, 412], "content": "\\gamma^{m a}", "score": 0.91, "index": 63}, {"type": "text", "coordinates": [196, 399, 393, 414], "content": " (where the superscript is taken mod ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [393, 404, 401, 410], "content": "\\kappa", "score": 0.84, "index": 65}, {"type": "text", "coordinates": [401, 399, 487, 414], "content": "), and stabilises ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [487, 400, 540, 413], "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "score": 0.93, "index": 67}, {"type": "inline_equation", "coordinates": [72, 415, 149, 427], "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "score": 0.89, "index": 68}, {"type": "text", "coordinates": [149, 412, 266, 429], "content": " iff the Jacobi symbol ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [266, 414, 286, 428], "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "score": 0.88, "index": 70}, {"type": "text", "coordinates": [287, 412, 326, 429], "content": " equals ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [327, 416, 342, 425], "content": "+1", "score": 0.54, "index": 72}, {"type": "text", "coordinates": [343, 412, 351, 429], "content": ").", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [94, 427, 133, 443], "content": "Why is", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [134, 430, 162, 438], "content": "k=2", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [163, 427, 491, 443], "content": " so special here? One reason is that rank-level duality associates", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [492, 430, 513, 442], "content": "B_{r,2}", "score": 0.95, "index": 77}, {"type": "text", "coordinates": [514, 427, 541, 443], "content": " with", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [71, 447, 115, 459], "content": "\\mathrm{u}(1)_{2r+1}", "score": 0.93, "index": 79}, {"type": "text", "coordinates": [116, 445, 281, 460], "content": ", and it is easy to confirm that", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [281, 442, 304, 459], "content": "\\widehat{\\mathrm{u(1)}}", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [304, 445, 541, 460], "content": " has a rich variety of fusion-symmetries (and", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [72, 460, 415, 474], "content": "modular invariants) coming from its si mple-currents. Also, the ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [416, 461, 438, 474], "content": "B_{r,2}", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [438, 460, 483, 474], "content": " matrix ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [483, 461, 491, 470], "content": "S", "score": 0.89, "index": 86}, {"type": "text", "coordinates": [492, 460, 540, 474], "content": " formally", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [70, 473, 427, 488], "content": "looks like the character table of the dihedral group and for some ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [427, 479, 434, 484], "content": "r", "score": 0.8, "index": 89}, {"type": "text", "coordinates": [434, 473, 541, 488], "content": " actually equals the", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [69, 487, 182, 504], "content": "Kac-Peterson matrix ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [183, 490, 191, 499], "content": "S", "score": 0.89, "index": 92}, {"type": "text", "coordinates": [191, 487, 363, 504], "content": " associated to the dihedral group", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [364, 488, 389, 504], "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [390, 487, 541, 504], "content": " twisted by an appropriate 3-", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [69, 503, 541, 518], "content": "cocycle [5] \u2014 finite group modular data tends to have significantly more modular invariants", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [70, 517, 349, 532], "content": "and fusion-symmetries than e.g. affine modular data.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [93, 537, 326, 556], "content": "Theorem 3.B. The fusion-symmetries of ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [327, 536, 350, 552], "content": "B_{r}^{(1)}", "score": 0.9, "index": 99}, {"type": "text", "coordinates": [351, 537, 383, 556], "content": "level ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [383, 539, 392, 551], "content": "k", "score": 0.71, "index": 101}, {"type": "text", "coordinates": [392, 537, 416, 556], "content": " for ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [416, 539, 453, 553], "content": "k\\ \\neq\\ 2", "score": 0.9, "index": 103}, {"type": "text", "coordinates": [453, 537, 479, 556], "content": " are ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [479, 539, 504, 553], "content": "\\pi[1]^{i}", "score": 0.91, "index": 105}, {"type": "text", "coordinates": [505, 537, 542, 556], "content": " where", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [71, 555, 120, 568], "content": "i\\in\\{0,1\\}", "score": 0.93, "index": 107}, {"type": "text", "coordinates": [120, 553, 151, 570], "content": ". For ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [151, 556, 181, 565], "content": "k=2", "score": 0.88, "index": 109}, {"type": "text", "coordinates": [181, 553, 338, 570], "content": " a fusion-symmetry will equal ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [339, 553, 395, 568], "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "score": 0.93, "index": 111}, {"type": "text", "coordinates": [396, 553, 417, 570], "content": " for ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [417, 554, 467, 568], "content": "i\\in\\{0,1\\}", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [468, 553, 493, 570], "content": " and ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [493, 554, 536, 568], "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [536, 553, 539, 570], "content": ",", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [71, 570, 126, 581], "content": "1\\leq m\\leq r", "score": 0.88, "index": 117}, {"type": "text", "coordinates": [126, 568, 131, 585], "content": ".", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [95, 587, 129, 606], "content": "When ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [130, 591, 159, 600], "content": "k=1", "score": 0.59, "index": 120}, {"type": "text", "coordinates": [159, 587, 165, 606], "content": ", ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [165, 589, 186, 603], "content": "\\pi[1]", "score": 0.41, "index": 122}, {"type": "text", "coordinates": [187, 587, 290, 606], "content": " is trivial. We have ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [290, 588, 449, 604], "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "score": 0.92, "index": 124}, {"type": "text", "coordinates": [68, 613, 160, 638], "content": "3.3. The algebra ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [161, 618, 183, 633], "content": "C_{r}^{(1)}", "score": 0.75, "index": 126}, {"type": "text", "coordinates": [183, 613, 190, 638], "content": ", ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [190, 623, 218, 633], "content": "r\\geq2", "score": 0.66, "index": 128}, {"type": "text", "coordinates": [96, 642, 144, 656], "content": "A weight ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [144, 643, 152, 653], "content": "\\lambda", "score": 0.82, "index": 130}, {"type": "text", "coordinates": [153, 642, 167, 656], "content": " of ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [167, 644, 183, 655], "content": "P_{+}", "score": 0.9, "index": 132}, {"type": "text", "coordinates": [183, 642, 230, 656], "content": " satisfies ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [230, 642, 339, 654], "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "score": 0.92, "index": 134}, {"type": "text", "coordinates": [339, 642, 364, 656], "content": " and ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [364, 642, 429, 653], "content": "\\kappa=k+r+1", "score": 0.9, "index": 136}, {"type": "text", "coordinates": [430, 642, 540, 656], "content": ". Charge-conjugation", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [71, 658, 81, 667], "content": "C", "score": 0.83, "index": 138}, {"type": "text", "coordinates": [81, 656, 325, 671], "content": " again is trivial, and there is a simple-current ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [325, 658, 334, 667], "content": "J", "score": 0.88, "index": 140}, {"type": "text", "coordinates": [334, 656, 396, 671], "content": " defined by ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [396, 657, 536, 670], "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "score": 0.9, "index": 142}, {"type": "text", "coordinates": [536, 656, 540, 671], "content": ",", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [70, 669, 98, 688], "content": "with ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [99, 671, 213, 686], "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "score": 0.92, "index": 145}, {"type": "text", "coordinates": [214, 669, 217, 688], "content": ".", "score": 1.0, "index": 146}, {"type": "text", "coordinates": [94, 683, 162, 702], "content": "Choose any ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [162, 686, 205, 698], "content": "\\lambda\\:\\in\\:P_{+}", "score": 0.88, "index": 148}, {"type": "text", "coordinates": [206, 683, 352, 702], "content": ". The Young diagram for ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [352, 686, 360, 695], "content": "\\lambda", "score": 0.83, "index": 150}, {"type": "text", "coordinates": [360, 683, 541, 702], "content": " is defined in the usual way: for", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [71, 704, 123, 715], "content": "1\\leq\\ell\\leq r", "score": 0.89, "index": 152}, {"type": "text", "coordinates": [124, 699, 151, 719], "content": ", the ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [151, 704, 157, 713], "content": "\\ell", "score": 0.54, "index": 154}, {"type": "text", "coordinates": [157, 699, 254, 719], "content": "th row consists of ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [254, 700, 335, 717], "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [336, 699, 400, 719], "content": " boxes. Let ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [401, 704, 415, 713], "content": "\\tau\\lambda", "score": 0.9, "index": 158}, {"type": "text", "coordinates": [415, 699, 479, 719], "content": " denote the ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [479, 704, 501, 717], "content": "C_{k,r}", "score": 0.93, "index": 160}, {"type": "text", "coordinates": [501, 699, 541, 719], "content": " weight", "score": 1.0, "index": 161}]	[]	[{"type": "block", "coordinates": [195, 246, 415, 285], "content": "", "caption": ""}, {"type": "block", "coordinates": [249, 309, 362, 349], "content": "", "caption": ""}, {"type": "inline", "coordinates": [161, 72, 183, 86], "content": "B_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [190, 77, 219, 87], "content": "r\\geq3", "caption": ""}, {"type": "inline", "coordinates": [145, 96, 154, 106], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [171, 96, 187, 109], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [234, 95, 429, 108], "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [459, 96, 537, 107], "content": "\\kappa=k+2r-1", "caption": ""}, {"type": "inline", "coordinates": [408, 109, 536, 123], "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "caption": ""}, {"type": "inline", "coordinates": [105, 123, 172, 138], "content": "Q(\\lambda)=\\lambda_{r}/2", "caption": ""}, {"type": "inline", "coordinates": [106, 209, 175, 221], "content": "1\\leq i<r-1", "caption": ""}, {"type": "inline", "coordinates": [181, 208, 210, 221], "content": "k>2", "caption": ""}, {"type": "inline", "coordinates": [240, 208, 290, 221], "content": "0<\\ell<k", "caption": ""}, {"type": "inline", "coordinates": [379, 208, 473, 222], "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [487, 209, 514, 219], "content": "\\ell=1", "caption": ""}, {"type": "inline", "coordinates": [106, 298, 203, 311], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "caption": ""}, {"type": "inline", "coordinates": [117, 358, 150, 367], "content": "k\\,=\\,2", "caption": ""}, {"type": "inline", "coordinates": [158, 357, 219, 368], "content": "\\kappa\\,=\\,2r+1", "caption": ""}, {"type": "inline", "coordinates": [441, 371, 521, 384], "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "caption": ""}, {"type": "inline", "coordinates": [71, 387, 166, 398], "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "caption": ""}, {"type": "inline", "coordinates": [198, 385, 293, 398], "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [379, 390, 389, 396], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "caption": ""}, {"type": "inline", "coordinates": [456, 390, 463, 396], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [471, 386, 501, 398], "content": "\\pi\\{m\\}", "caption": ""}, {"type": "inline", "coordinates": [533, 387, 540, 396], "content": "0", "caption": ""}, {"type": "inline", "coordinates": [95, 401, 103, 410], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [142, 401, 155, 412], "content": "\\gamma^{a}", "caption": ""}, {"type": "inline", "coordinates": [174, 401, 195, 412], "content": "\\gamma^{m a}", "caption": ""}, {"type": "inline", "coordinates": [393, 404, 401, 410], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [487, 400, 540, 413], "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "caption": ""}, {"type": "inline", "coordinates": [72, 415, 149, 427], "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "caption": ""}, 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"caption": ""}, {"type": "inline", "coordinates": [71, 570, 126, 581], "content": "1\\leq m\\leq r", "caption": ""}, {"type": "inline", "coordinates": [130, 591, 159, 600], "content": "k=1", "caption": ""}, {"type": "inline", "coordinates": [165, 589, 186, 603], "content": "\\pi[1]", "caption": ""}, {"type": "inline", "coordinates": [290, 588, 449, 604], "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "caption": ""}, {"type": "inline", "coordinates": [161, 618, 183, 633], "content": "C_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [190, 623, 218, 633], "content": "r\\geq2", "caption": ""}, {"type": "inline", "coordinates": [144, 643, 152, 653], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [167, 644, 183, 655], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [230, 642, 339, 654], "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [364, 642, 429, 653], "content": "\\kappa=k+r+1", "caption": ""}, {"type": "inline", "coordinates": [71, 658, 81, 667], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [325, 658, 334, 667], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [396, 657, 536, 670], "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "caption": ""}, {"type": "inline", "coordinates": [99, 671, 213, 686], "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [162, 686, 205, 698], "content": "\\lambda\\:\\in\\:P_{+}", "caption": ""}, {"type": "inline", "coordinates": [352, 686, 360, 695], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [71, 704, 123, 715], "content": "1\\leq\\ell\\leq r", "caption": ""}, {"type": "inline", "coordinates": [151, 704, 157, 713], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [254, 700, 335, 717], "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [401, 704, 415, 713], "content": "\\tau\\lambda", "caption": ""}, {"type": "inline", "coordinates": [479, 704, 501, 717], "content": "C_{k,r}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "3.2. The algebra $B_{r}^{(1)}$ , $r\\geq3$ ", "page_idx": 9}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}$ , and $\\kappa=k+2r-1$ . The charge-conjugation is trivial, but there is a simple-current: $J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)$ . It has $Q(\\lambda)=\\lambda_{r}/2$ . ", "page_idx": 9}, {"type": "text", "text": "The only fusion products we need are ", "page_idx": 9}, {"type": "text", "text": "for all $1\\leq i<r-1$ , $k>2$ , and $0<\\ell<k$ , where we drop $\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}$ if $\\ell=1$ . We will also use the character formula (2.1b) ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$ and ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "For $k\\,=\\,2$ ( $\\kappa\\,=\\,2r+1$ ) there are several Galois fusion-symmetries \u2014 one for each Galois automorphism, since S020 = 41\u03ba is rational. In particular, define $\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}$ for $i\\,=\\,1,2,\\dots,r\\,-\\,1$ , and $\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}$ . Then for any ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ coprime to $\\kappa$ , $\\pi\\{m\\}$ fixes $0$ and $J$ , sends $\\gamma^{a}$ to $\\gamma^{m a}$ (where the superscript is taken mod $\\kappa$ ), and stabilises $\\{\\Lambda_{r},J\\Lambda_{r}\\}$ $(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}$ iff the Jacobi symbol $\\Bigl(\\frac{\\kappa}{m}\\Bigr)$ equals $+1$ ). ", "page_idx": 9}, {"type": "text", "text": "Why is $k=2$ so special here? One reason is that rank-level duality associates $B_{r,2}$ with $\\mathrm{u}(1)_{2r+1}$ , and it is easy to confirm that $\\widehat{\\mathrm{u(1)}}$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $B_{r,2}$ matrix $S$ formally looks like the character table of the dihedral group and for some $r$ actually equals the Kac-Peterson matrix $S$ associated to the dihedral group ${\\mathfrak{D}}_{{\\sqrt{\\kappa}}}$ twisted by an appropriate 3- cocycle [5] \u2014 finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data. ", "page_idx": 9}, {"type": "text", "text": "Theorem 3.B. The fusion-symmetries of $B_{r}^{(1)}$ level $k$ for $k\\ \\neq\\ 2$ are $\\pi[1]^{i}$ where $i\\in\\{0,1\\}$ . For $k=2$ a fusion-symmetry will equal $\\pi[1]^{i}\\,\\pi\\{m\\}$ for $i\\in\\{0,1\\}$ and $m\\in\\mathbb{Z}_{\\kappa}^{\\times}$ , $1\\leq m\\leq r$ . ", "page_idx": 9}, {"type": "text", "text": "When $k=1$ , $\\pi[1]$ is trivial. We have $\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).$ ", "page_idx": 9}, {"type": "text", "text": "3.3. The algebra $C_{r}^{(1)}$ , $r\\geq2$ ", "page_idx": 9}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}$ and $\\kappa=k+r+1$ . Charge-conjugation $C$ again is trivial, and there is a simple-current $J$ defined by $J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)$ , with $\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}$ . ", "page_idx": 9}, {"type": "text", "text": "Choose any $\\lambda\\:\\in\\:P_{+}$ . The Young diagram for $\\lambda$ is defined in the usual way: for $1\\leq\\ell\\leq r$ , the $\\ell$ th row consists of $\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}$ boxes. Let $\\tau\\lambda$ denote the $C_{k,r}$ weight whose diagram is the transpose of that for $\\lambda$ . (For this purpose the algebra $C_{1}$ may be identified with $A_{1}$ .) For example, $\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}$ , where we use tilde\u2019s to denote the quantities of $C_{k,r}$ . In fact, $\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})$ is a bijection. 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We", "type": "text"}], "index": 5}, {"bbox": [71, 223, 288, 238], "spans": [{"bbox": [71, 223, 288, 238], "score": 1.0, "content": "will also use the character formula (2.1b)", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 295, 227, 310], "lines": [{"bbox": [72, 297, 227, 312], "spans": [{"bbox": [72, 297, 105, 312], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 298, 203, 311], "score": 0.94, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [203, 297, 227, 312], "score": 1.0, "content": " and", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 353, 541, 426], "lines": [{"bbox": [93, 354, 541, 371], "spans": [{"bbox": [93, 354, 117, 371], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 358, 150, 367], "score": 0.89, "content": "k\\,=\\,2", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [150, 354, 158, 371], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [158, 357, 219, 368], "score": 0.69, "content": "\\kappa\\,=\\,2r+1", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [220, 354, 541, 371], "score": 1.0, "content": ") there are several Galois fusion-symmetries \u2014 one for each", "type": "text"}], "index": 10}, {"bbox": [69, 368, 541, 388], "spans": [{"bbox": [69, 368, 272, 388], "score": 1.0, "content": "Galois automorphism, since S020 = 41\u03ba ", "type": "text"}, {"bbox": [267, 369, 441, 387], "score": 1.0, "content": "is rational. In particular, define ", "type": "text"}, {"bbox": [441, 371, 521, 384], "score": 0.94, "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [521, 369, 541, 387], "score": 1.0, "content": " for", "type": "text"}], "index": 11}, {"bbox": [71, 383, 540, 402], "spans": [{"bbox": [71, 387, 166, 398], "score": 0.9, "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [166, 383, 198, 402], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [198, 385, 293, 398], "score": 0.92, "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [293, 383, 378, 402], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [379, 390, 389, 396], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [390, 383, 455, 402], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [456, 390, 463, 396], "score": 0.69, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [463, 383, 470, 402], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [471, 386, 501, 398], "score": 0.93, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [501, 383, 533, 402], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [533, 387, 540, 396], "score": 0.39, "content": "0", "type": "inline_equation", "height": 9, "width": 7}], "index": 12}, {"bbox": [71, 399, 540, 414], "spans": [{"bbox": [71, 399, 95, 414], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 401, 103, 410], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [103, 399, 142, 414], "score": 1.0, "content": ", sends ", "type": "text"}, {"bbox": [142, 401, 155, 412], "score": 0.91, "content": "\\gamma^{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [155, 399, 173, 414], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 401, 195, 412], "score": 0.91, "content": "\\gamma^{m a}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [196, 399, 393, 414], "score": 1.0, "content": " (where the superscript is taken mod ", "type": "text"}, {"bbox": [393, 404, 401, 410], "score": 0.84, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [401, 399, 487, 414], "score": 1.0, "content": "), and stabilises ", "type": "text"}, {"bbox": [487, 400, 540, 413], "score": 0.93, "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 53}], "index": 13}, {"bbox": [72, 412, 351, 429], "spans": [{"bbox": [72, 415, 149, 427], "score": 0.89, "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [149, 412, 266, 429], "score": 1.0, "content": " iff the Jacobi symbol ", "type": "text"}, {"bbox": [266, 414, 286, 428], "score": 0.88, "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [287, 412, 326, 429], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [327, 416, 342, 425], "score": 0.54, "content": "+1", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [343, 412, 351, 429], "score": 1.0, "content": ").", "type": "text"}], "index": 14}], "index": 12}, {"type": "text", "bbox": [70, 426, 541, 530], "lines": [{"bbox": [94, 427, 541, 443], "spans": [{"bbox": [94, 427, 133, 443], "score": 1.0, "content": "Why is", "type": "text"}, {"bbox": [134, 430, 162, 438], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [163, 427, 491, 443], "score": 1.0, "content": " so special here? One reason is that rank-level duality associates", "type": "text"}, {"bbox": [492, 430, 513, 442], "score": 0.95, "content": "B_{r,2}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [514, 427, 541, 443], "score": 1.0, "content": " with", "type": "text"}], "index": 15}, {"bbox": [71, 442, 541, 460], "spans": [{"bbox": [71, 447, 115, 459], "score": 0.93, "content": "\\mathrm{u}(1)_{2r+1}", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [116, 445, 281, 460], "score": 1.0, "content": ", and it is easy to confirm that", "type": "text"}, {"bbox": [281, 442, 304, 459], "score": 0.9, "content": "\\widehat{\\mathrm{u(1)}}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [304, 445, 541, 460], "score": 1.0, "content": " has a rich variety of fusion-symmetries (and", "type": "text"}], "index": 16}, {"bbox": [72, 460, 540, 474], "spans": [{"bbox": [72, 460, 415, 474], "score": 1.0, "content": "modular invariants) coming from its si mple-currents. Also, the ", "type": "text"}, {"bbox": [416, 461, 438, 474], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [438, 460, 483, 474], "score": 1.0, "content": " matrix ", "type": "text"}, {"bbox": [483, 461, 491, 470], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [492, 460, 540, 474], "score": 1.0, "content": " formally", "type": "text"}], "index": 17}, {"bbox": [70, 473, 541, 488], "spans": [{"bbox": [70, 473, 427, 488], "score": 1.0, "content": "looks like the character table of the dihedral group and for some ", "type": "text"}, {"bbox": [427, 479, 434, 484], "score": 0.8, "content": "r", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [434, 473, 541, 488], "score": 1.0, "content": " actually equals the", "type": "text"}], "index": 18}, {"bbox": [69, 487, 541, 504], "spans": [{"bbox": [69, 487, 182, 504], "score": 1.0, "content": "Kac-Peterson matrix ", "type": "text"}, {"bbox": [183, 490, 191, 499], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [191, 487, 363, 504], "score": 1.0, "content": " associated to the dihedral group", "type": "text"}, {"bbox": [364, 488, 389, 504], "score": 0.91, "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [390, 487, 541, 504], "score": 1.0, "content": " twisted by an appropriate 3-", "type": "text"}], "index": 19}, {"bbox": [69, 503, 541, 518], "spans": [{"bbox": [69, 503, 541, 518], "score": 1.0, "content": "cocycle [5] \u2014 finite group modular data tends to have significantly more modular invariants", "type": "text"}], "index": 20}, {"bbox": [70, 517, 349, 532], "spans": [{"bbox": [70, 517, 349, 532], "score": 1.0, "content": "and fusion-symmetries than e.g. affine modular data.", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [70, 535, 542, 581], "lines": [{"bbox": [93, 536, 542, 556], "spans": [{"bbox": [93, 537, 326, 556], "score": 1.0, "content": "Theorem 3.B. The fusion-symmetries of ", "type": "text"}, {"bbox": [327, 536, 350, 552], "score": 0.9, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [351, 537, 383, 556], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [383, 539, 392, 551], "score": 0.71, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [392, 537, 416, 556], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [416, 539, 453, 553], "score": 0.9, "content": "k\\ \\neq\\ 2", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [453, 537, 479, 556], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [479, 539, 504, 553], "score": 0.91, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [505, 537, 542, 556], "score": 1.0, "content": " where", "type": "text"}], "index": 22}, {"bbox": [71, 553, 539, 570], "spans": [{"bbox": [71, 555, 120, 568], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [120, 553, 151, 570], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [151, 556, 181, 565], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [181, 553, 338, 570], "score": 1.0, "content": " a fusion-symmetry will equal ", "type": "text"}, {"bbox": [339, 553, 395, 568], "score": 0.93, "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [396, 553, 417, 570], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [417, 554, 467, 568], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [468, 553, 493, 570], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 554, 536, 568], "score": 0.91, "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [536, 553, 539, 570], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [71, 568, 131, 585], "spans": [{"bbox": [71, 570, 126, 581], "score": 0.88, "content": "1\\leq m\\leq r", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [126, 568, 131, 585], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23}, {"type": "text", "bbox": [93, 586, 451, 604], "lines": [{"bbox": [95, 587, 449, 606], "spans": [{"bbox": [95, 587, 129, 606], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 591, 159, 600], "score": 0.59, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [159, 587, 165, 606], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [165, 589, 186, 603], "score": 0.41, "content": "\\pi[1]", "type": "inline_equation", "height": 14, "width": 21}, {"bbox": [187, 587, 290, 606], "score": 1.0, "content": " is trivial. We have ", "type": "text"}, {"bbox": [290, 588, 449, 604], "score": 0.92, "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "type": "inline_equation", "height": 16, "width": 159}], "index": 25}], "index": 25}, {"type": "text", "bbox": [71, 616, 220, 633], "lines": [{"bbox": [68, 613, 218, 638], "spans": [{"bbox": [68, 613, 160, 638], "score": 1.0, "content": "3.3. The algebra ", "type": "text"}, {"bbox": [161, 618, 183, 633], "score": 0.75, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [183, 613, 190, 638], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 623, 218, 633], "score": 0.66, "content": "r\\geq2", "type": "inline_equation", "height": 10, "width": 28}], "index": 26}], "index": 26}, {"type": "text", "bbox": [71, 639, 541, 683], "lines": [{"bbox": [96, 642, 540, 656], "spans": [{"bbox": [96, 642, 144, 656], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 643, 152, 653], "score": 0.82, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [153, 642, 167, 656], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [167, 644, 183, 655], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [183, 642, 230, 656], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [230, 642, 339, 654], "score": 0.92, "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [339, 642, 364, 656], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [364, 642, 429, 653], "score": 0.9, "content": "\\kappa=k+r+1", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [430, 642, 540, 656], "score": 1.0, "content": ". Charge-conjugation", "type": "text"}], "index": 27}, {"bbox": [71, 656, 540, 671], "spans": [{"bbox": [71, 658, 81, 667], "score": 0.83, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [81, 656, 325, 671], "score": 1.0, "content": " again is trivial, and there is a simple-current ", "type": "text"}, {"bbox": [325, 658, 334, 667], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [334, 656, 396, 671], "score": 1.0, "content": " defined by ", "type": "text"}, {"bbox": [396, 657, 536, 670], "score": 0.9, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "type": "inline_equation", "height": 13, "width": 140}, {"bbox": [536, 656, 540, 671], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [70, 669, 217, 688], "spans": [{"bbox": [70, 669, 98, 688], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 671, 213, 686], "score": 0.92, "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "type": "inline_equation", "height": 15, "width": 114}, {"bbox": [214, 669, 217, 688], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28}, {"type": "text", "bbox": [70, 683, 540, 716], "lines": [{"bbox": [94, 683, 541, 702], "spans": [{"bbox": [94, 683, 162, 702], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [162, 686, 205, 698], "score": 0.88, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [206, 683, 352, 702], "score": 1.0, "content": ". The Young diagram for ", "type": "text"}, {"bbox": [352, 686, 360, 695], "score": 0.83, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 683, 541, 702], "score": 1.0, "content": " is defined in the usual way: for", "type": "text"}], "index": 30}, {"bbox": [71, 699, 541, 719], "spans": [{"bbox": [71, 704, 123, 715], "score": 0.89, "content": "1\\leq\\ell\\leq r", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [124, 699, 151, 719], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [151, 704, 157, 713], "score": 0.54, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [157, 699, 254, 719], "score": 1.0, "content": "th row consists of ", "type": "text"}, {"bbox": [254, 700, 335, 717], "score": 0.93, "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "type": "inline_equation", "height": 17, "width": 81}, {"bbox": [336, 699, 400, 719], "score": 1.0, "content": " boxes. Let ", "type": "text"}, {"bbox": [401, 704, 415, 713], "score": 0.9, "content": "\\tau\\lambda", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [415, 699, 479, 719], "score": 1.0, "content": " denote the ", "type": "text"}, {"bbox": [479, 704, 501, 717], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [501, 699, 541, 719], "score": 1.0, "content": " weight", "type": "text"}], "index": 31}], "index": 30.5}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 313, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 69, 221, 86], "lines": [{"bbox": [68, 67, 219, 92], "spans": [{"bbox": [68, 67, 160, 92], "score": 1.0, "content": "3.2. The algebra ", "type": "text"}, {"bbox": [161, 72, 183, 86], "score": 0.73, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [184, 67, 190, 92], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 77, 219, 87], "score": 0.74, "content": "r\\geq3", "type": "inline_equation", "height": 10, "width": 29}], "index": 0}], "index": 0, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [68, 67, 219, 92]}, {"type": "text", "bbox": [70, 92, 542, 136], "lines": [{"bbox": [95, 95, 541, 110], "spans": [{"bbox": [95, 95, 145, 110], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [145, 96, 154, 106], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [154, 95, 170, 110], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [171, 96, 187, 109], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [187, 95, 233, 110], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [234, 95, 429, 108], "score": 0.91, "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "type": "inline_equation", "height": 13, "width": 195}, {"bbox": [430, 95, 459, 110], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [459, 96, 537, 107], "score": 0.9, "content": "\\kappa=k+2r-1", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [537, 95, 541, 110], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [70, 109, 541, 125], "spans": [{"bbox": [70, 109, 408, 125], "score": 1.0, "content": "The charge-conjugation is trivial, but there is a simple-current: ", "type": "text"}, {"bbox": [408, 109, 536, 123], "score": 0.91, "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [536, 109, 541, 125], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [69, 123, 176, 140], "spans": [{"bbox": [69, 123, 105, 140], "score": 1.0, "content": "It has ", "type": "text"}, {"bbox": [105, 123, 172, 138], "score": 0.92, "content": "Q(\\lambda)=\\lambda_{r}/2", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [172, 123, 176, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [69, 95, 541, 140]}, {"type": "text", "bbox": [94, 137, 294, 151], "lines": [{"bbox": [95, 138, 294, 152], "spans": [{"bbox": [95, 138, 294, 152], "score": 1.0, "content": "The only fusion products we need are", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [95, 138, 294, 152]}, {"type": "text", "bbox": [69, 205, 541, 235], "lines": [{"bbox": [70, 208, 541, 223], "spans": [{"bbox": [70, 208, 105, 223], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 209, 175, 221], "score": 0.83, "content": "1\\leq i<r-1", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [175, 208, 180, 223], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [181, 208, 210, 221], "score": 0.68, "content": "k>2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [211, 208, 240, 223], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [240, 208, 290, 221], "score": 0.91, "content": "0<\\ell<k", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [291, 208, 378, 223], "score": 1.0, "content": ", where we drop ", "type": "text"}, {"bbox": [379, 208, 473, 222], "score": 0.91, "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [473, 208, 487, 223], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [487, 209, 514, 219], "score": 0.87, "content": "\\ell=1", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [515, 208, 541, 223], "score": 1.0, "content": ". We", "type": "text"}], "index": 5}, {"bbox": [71, 223, 288, 238], "spans": [{"bbox": [71, 223, 288, 238], "score": 1.0, "content": "will also use the character formula (2.1b)", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [70, 208, 541, 238]}, {"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 295, 227, 310], "lines": [{"bbox": [72, 297, 227, 312], "spans": [{"bbox": [72, 297, 105, 312], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 298, 203, 311], "score": 0.94, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [203, 297, 227, 312], "score": 1.0, "content": " and", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [72, 297, 227, 312]}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 353, 541, 426], "lines": [{"bbox": [93, 354, 541, 371], "spans": [{"bbox": [93, 354, 117, 371], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 358, 150, 367], "score": 0.89, "content": "k\\,=\\,2", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [150, 354, 158, 371], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [158, 357, 219, 368], "score": 0.69, "content": "\\kappa\\,=\\,2r+1", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [220, 354, 541, 371], "score": 1.0, "content": ") there are several Galois fusion-symmetries \u2014 one for each", "type": "text"}], "index": 10}, {"bbox": [69, 368, 541, 388], "spans": [{"bbox": [69, 368, 272, 388], "score": 1.0, "content": "Galois automorphism, since S020 = 41\u03ba ", "type": "text"}, {"bbox": [267, 369, 441, 387], "score": 1.0, "content": "is rational. In particular, define ", "type": "text"}, {"bbox": [441, 371, 521, 384], "score": 0.94, "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [521, 369, 541, 387], "score": 1.0, "content": " for", "type": "text"}], "index": 11}, {"bbox": [71, 383, 540, 402], "spans": [{"bbox": [71, 387, 166, 398], "score": 0.9, "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [166, 383, 198, 402], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [198, 385, 293, 398], "score": 0.92, "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [293, 383, 378, 402], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [379, 390, 389, 396], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [390, 383, 455, 402], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [456, 390, 463, 396], "score": 0.69, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [463, 383, 470, 402], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [471, 386, 501, 398], "score": 0.93, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [501, 383, 533, 402], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [533, 387, 540, 396], "score": 0.39, "content": "0", "type": "inline_equation", "height": 9, "width": 7}], "index": 12}, {"bbox": [71, 399, 540, 414], "spans": [{"bbox": [71, 399, 95, 414], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 401, 103, 410], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [103, 399, 142, 414], "score": 1.0, "content": ", sends ", "type": "text"}, {"bbox": [142, 401, 155, 412], "score": 0.91, "content": "\\gamma^{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [155, 399, 173, 414], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 401, 195, 412], "score": 0.91, "content": "\\gamma^{m a}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [196, 399, 393, 414], "score": 1.0, "content": " (where the superscript is taken mod ", "type": "text"}, {"bbox": [393, 404, 401, 410], "score": 0.84, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [401, 399, 487, 414], "score": 1.0, "content": "), and stabilises ", "type": "text"}, {"bbox": [487, 400, 540, 413], "score": 0.93, "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 53}], "index": 13}, {"bbox": [72, 412, 351, 429], "spans": [{"bbox": [72, 415, 149, 427], "score": 0.89, "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [149, 412, 266, 429], "score": 1.0, "content": " iff the Jacobi symbol ", "type": "text"}, {"bbox": [266, 414, 286, 428], "score": 0.88, "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [287, 412, 326, 429], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [327, 416, 342, 425], "score": 0.54, "content": "+1", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [343, 412, 351, 429], "score": 1.0, "content": ").", "type": "text"}], "index": 14}], "index": 12, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [69, 354, 541, 429]}, {"type": "text", "bbox": [70, 426, 541, 530], "lines": [{"bbox": [94, 427, 541, 443], "spans": [{"bbox": [94, 427, 133, 443], "score": 1.0, "content": "Why is", "type": "text"}, {"bbox": [134, 430, 162, 438], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [163, 427, 491, 443], "score": 1.0, "content": " so special here? One reason is that rank-level duality associates", "type": "text"}, {"bbox": [492, 430, 513, 442], "score": 0.95, "content": "B_{r,2}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [514, 427, 541, 443], "score": 1.0, "content": " with", "type": "text"}], "index": 15}, {"bbox": [71, 442, 541, 460], "spans": [{"bbox": [71, 447, 115, 459], "score": 0.93, "content": "\\mathrm{u}(1)_{2r+1}", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [116, 445, 281, 460], "score": 1.0, "content": ", and it is easy to confirm that", "type": "text"}, {"bbox": [281, 442, 304, 459], "score": 0.9, "content": "\\widehat{\\mathrm{u(1)}}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [304, 445, 541, 460], "score": 1.0, "content": " has a rich variety of fusion-symmetries (and", "type": "text"}], "index": 16}, {"bbox": [72, 460, 540, 474], "spans": [{"bbox": [72, 460, 415, 474], "score": 1.0, "content": "modular invariants) coming from its si mple-currents. Also, the ", "type": "text"}, {"bbox": [416, 461, 438, 474], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [438, 460, 483, 474], "score": 1.0, "content": " matrix ", "type": "text"}, {"bbox": [483, 461, 491, 470], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [492, 460, 540, 474], "score": 1.0, "content": " formally", "type": "text"}], "index": 17}, {"bbox": [70, 473, 541, 488], "spans": [{"bbox": [70, 473, 427, 488], "score": 1.0, "content": "looks like the character table of the dihedral group and for some ", "type": "text"}, {"bbox": [427, 479, 434, 484], "score": 0.8, "content": "r", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [434, 473, 541, 488], "score": 1.0, "content": " actually equals the", "type": "text"}], "index": 18}, {"bbox": [69, 487, 541, 504], "spans": [{"bbox": [69, 487, 182, 504], "score": 1.0, "content": "Kac-Peterson matrix ", "type": "text"}, {"bbox": [183, 490, 191, 499], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [191, 487, 363, 504], "score": 1.0, "content": " associated to the dihedral group", "type": "text"}, {"bbox": [364, 488, 389, 504], "score": 0.91, "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [390, 487, 541, 504], "score": 1.0, "content": " twisted by an appropriate 3-", "type": "text"}], "index": 19}, {"bbox": [69, 503, 541, 518], "spans": [{"bbox": [69, 503, 541, 518], "score": 1.0, "content": "cocycle [5] \u2014 finite group modular data tends to have significantly more modular invariants", "type": "text"}], "index": 20}, {"bbox": [70, 517, 349, 532], "spans": [{"bbox": [70, 517, 349, 532], "score": 1.0, "content": "and fusion-symmetries than e.g. affine modular data.", "type": "text"}], "index": 21}], "index": 18, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [69, 427, 541, 532]}, {"type": "text", "bbox": [70, 535, 542, 581], "lines": [{"bbox": [93, 536, 542, 556], "spans": [{"bbox": [93, 537, 326, 556], "score": 1.0, "content": "Theorem 3.B. The fusion-symmetries of ", "type": "text"}, {"bbox": [327, 536, 350, 552], "score": 0.9, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [351, 537, 383, 556], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [383, 539, 392, 551], "score": 0.71, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [392, 537, 416, 556], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [416, 539, 453, 553], "score": 0.9, "content": "k\\ \\neq\\ 2", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [453, 537, 479, 556], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [479, 539, 504, 553], "score": 0.91, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [505, 537, 542, 556], "score": 1.0, "content": " where", "type": "text"}], "index": 22}, {"bbox": [71, 553, 539, 570], "spans": [{"bbox": [71, 555, 120, 568], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [120, 553, 151, 570], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [151, 556, 181, 565], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [181, 553, 338, 570], "score": 1.0, "content": " a fusion-symmetry will equal ", "type": "text"}, {"bbox": [339, 553, 395, 568], "score": 0.93, "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [396, 553, 417, 570], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [417, 554, 467, 568], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [468, 553, 493, 570], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 554, 536, 568], "score": 0.91, "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [536, 553, 539, 570], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [71, 568, 131, 585], "spans": [{"bbox": [71, 570, 126, 581], "score": 0.88, "content": "1\\leq m\\leq r", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [126, 568, 131, 585], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [71, 536, 542, 585]}, {"type": "text", "bbox": [93, 586, 451, 604], "lines": [{"bbox": [95, 587, 449, 606], "spans": [{"bbox": [95, 587, 129, 606], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 591, 159, 600], "score": 0.59, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [159, 587, 165, 606], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [165, 589, 186, 603], "score": 0.41, "content": "\\pi[1]", "type": "inline_equation", "height": 14, "width": 21}, {"bbox": [187, 587, 290, 606], "score": 1.0, "content": " is trivial. We have ", "type": "text"}, {"bbox": [290, 588, 449, 604], "score": 0.92, "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "type": "inline_equation", "height": 16, "width": 159}], "index": 25}], "index": 25, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [95, 587, 449, 606]}, {"type": "text", "bbox": [71, 616, 220, 633], "lines": [{"bbox": [68, 613, 218, 638], "spans": [{"bbox": [68, 613, 160, 638], "score": 1.0, "content": "3.3. The algebra ", "type": "text"}, {"bbox": [161, 618, 183, 633], "score": 0.75, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [183, 613, 190, 638], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 623, 218, 633], "score": 0.66, "content": "r\\geq2", "type": "inline_equation", "height": 10, "width": 28}], "index": 26}], "index": 26, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [68, 613, 218, 638]}, {"type": "text", "bbox": [71, 639, 541, 683], "lines": [{"bbox": [96, 642, 540, 656], "spans": [{"bbox": [96, 642, 144, 656], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 643, 152, 653], "score": 0.82, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [153, 642, 167, 656], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [167, 644, 183, 655], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [183, 642, 230, 656], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [230, 642, 339, 654], "score": 0.92, "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [339, 642, 364, 656], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [364, 642, 429, 653], "score": 0.9, "content": "\\kappa=k+r+1", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [430, 642, 540, 656], "score": 1.0, "content": ". Charge-conjugation", "type": "text"}], "index": 27}, {"bbox": [71, 656, 540, 671], "spans": [{"bbox": [71, 658, 81, 667], "score": 0.83, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [81, 656, 325, 671], "score": 1.0, "content": " again is trivial, and there is a simple-current ", "type": "text"}, {"bbox": [325, 658, 334, 667], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [334, 656, 396, 671], "score": 1.0, "content": " defined by ", "type": "text"}, {"bbox": [396, 657, 536, 670], "score": 0.9, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "type": "inline_equation", "height": 13, "width": 140}, {"bbox": [536, 656, 540, 671], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [70, 669, 217, 688], "spans": [{"bbox": [70, 669, 98, 688], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 671, 213, 686], "score": 0.92, "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "type": "inline_equation", "height": 15, "width": 114}, {"bbox": [214, 669, 217, 688], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [70, 642, 540, 688]}, {"type": "text", "bbox": [70, 683, 540, 716], "lines": [{"bbox": [94, 683, 541, 702], "spans": [{"bbox": [94, 683, 162, 702], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [162, 686, 205, 698], "score": 0.88, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [206, 683, 352, 702], "score": 1.0, "content": ". The Young diagram for ", "type": "text"}, {"bbox": [352, 686, 360, 695], "score": 0.83, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 683, 541, 702], "score": 1.0, "content": " is defined in the usual way: for", "type": "text"}], "index": 30}, {"bbox": [71, 699, 541, 719], "spans": [{"bbox": [71, 704, 123, 715], "score": 0.89, "content": "1\\leq\\ell\\leq r", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [124, 699, 151, 719], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [151, 704, 157, 713], "score": 0.54, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [157, 699, 254, 719], "score": 1.0, "content": "th row consists of ", "type": "text"}, {"bbox": [254, 700, 335, 717], "score": 0.93, "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "type": "inline_equation", "height": 17, "width": 81}, {"bbox": [336, 699, 400, 719], "score": 1.0, "content": " boxes. Let ", "type": "text"}, {"bbox": [401, 704, 415, 713], "score": 0.9, "content": "\\tau\\lambda", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [415, 699, 479, 719], "score": 1.0, "content": " denote the ", "type": "text"}, {"bbox": [479, 704, 501, 717], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [501, 699, 541, 719], "score": 1.0, "content": " weight", "type": "text"}], "index": 31}, {"bbox": [72, 74, 541, 88], "spans": [{"bbox": [72, 74, 303, 88], "score": 1.0, "content": "whose diagram is the transpose of that for ", "type": "text", "cross_page": true}, {"bbox": [303, 75, 311, 84], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [311, 74, 481, 88], "score": 1.0, "content": ". (For this purpose the algebra ", "type": "text", "cross_page": true}, {"bbox": [482, 75, 495, 86], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [496, 74, 541, 88], "score": 1.0, "content": " may be", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 87, 541, 103], "spans": [{"bbox": [70, 87, 149, 103], "score": 1.0, "content": "identified with ", "type": "text", "cross_page": true}, {"bbox": [149, 90, 164, 101], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [164, 87, 246, 103], "score": 1.0, "content": ".) For example,", "type": "text", "cross_page": true}, {"bbox": [247, 87, 304, 101], "score": 0.94, "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 57, "cross_page": true}, {"bbox": [304, 87, 541, 103], "score": 1.0, "content": ", where we use tilde\u2019s to denote the quantities", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [70, 102, 394, 117], "spans": [{"bbox": [70, 102, 84, 117], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [85, 104, 107, 117], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22, "cross_page": true}, {"bbox": [107, 102, 157, 117], "score": 1.0, "content": ". In fact, ", "type": "text", "cross_page": true}, {"bbox": [157, 103, 286, 117], "score": 0.94, "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 129, "cross_page": true}, {"bbox": [286, 102, 394, 117], "score": 1.0, "content": " is a bijection. Then", "type": "text", "cross_page": true}], "index": 2}], "index": 30.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [71, 683, 541, 719]}]}
0002044v1	14	There are Galois fusion-symmetries at levels $$k=3$$ and 4. In particular, for $$k=3$$ we have the fusion-symmetry $$\pi_{3}=\pi\{5\}$$ which interchanges both $$\Lambda_{2}\leftrightarrow\Lambda_{4}$$ and $$\Lambda_{1}\leftrightarrow3\Lambda_{4}$$ , and fixes the other five weights in $$P_{+}$$ . The exceptional $$\pi_{3}$$ was found independently in [34,14]. For $$k\,=\,4$$ we get a fusion-symmetry of order 4, which we will call $$\pi_{4}$$ . It fixes 0, $$\Lambda_{2}+\Lambda_{4}$$ , $$\Lambda_{3}+\Lambda_{4}$$ , and $$2\Lambda_{4}$$ , and permutes $$\Lambda_{4}\;\mapsto\;\Lambda_{1}\;\mapsto\;2\Lambda_{1}\;\mapsto\;4\Lambda_{4}\;\mapsto\;\Lambda_{4}$$ , $$\Lambda_{2}\mapsto2\Lambda_{3}\mapsto3\Lambda_{4}\mapsto\Lambda_{3}\mapsto\Lambda_{2}$$ , and $$\Lambda_{1}\!+\!\Lambda_{3}\mapsto\Lambda_{3}\!+\!2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{4}\mapsto\Lambda_{1}+2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{3}$$ . Its square $$\pi_{4}^{2}$$ equals the fusion-symmetry $$\pi\{5\}$$ . The only fusion products we need can be obtained from [29] and (2.4): $$\Lambda_{4}$$ × $$\Lambda_{4}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{1}$$ + $$(2\Lambda_{4})_{2}$$ $$\Lambda_{1}$$ × $$\Lambda_{4}=(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ $$\Lambda_{3}$$ × $$\Lambda_{4}=(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{3}$$ + $$(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ + $$(\Lambda_{3}+\Lambda_{4})_{3}$$ + $$(2\Lambda_{4})_{2}$$ $$(2\Lambda_{4})$$ × $$\Lambda_{4}=(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(2\Lambda_{4})_{2}$$ + $$(3\Lambda_{4})_{3}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ + $$(\Lambda_{3}+\Lambda_{4})_{3}$$ Theorem 3.F4. The only nontrivial fusion-symmetries of $${F}_{4}^{(1)}$$ are $$\pi_{3}$$ at level 3, and $$\pi_{4}^{i}$$ for $$1\leq i\leq3$$ , which occur at level 4. # 3.9. The algebra $${G_{2}^{(1)}}$$ A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+\lambda_{2}$$ , and $$\kappa=k+4$$ . The conjugations and simple-currents are all trivial. Again there are nontrivial Galois fusion-symmetries. At $$k=3$$ , we have the order 3 fusion-symmetry $$\pi_{3}=\pi\{4\}$$ sending $$\Lambda_{1}\mapsto3\Lambda_{2}\mapsto\Lambda_{2}\mapsto\Lambda_{1}$$ , and fixing the remaining three weights. It was found in [14]. At $$k=4$$ , we have $$\pi_{4}=\pi\{5\}$$ permuting both $$\Lambda_{1}\leftrightarrow4\Lambda_{2}$$ and $$2\Lambda_{1}\leftrightarrow\Lambda_{2}$$ , and fixing the other five weights. It was found independently in [34,14], and in $$\S5$$ we will see that it is closely related to the $$\pi_{3}$$ of $$F_{4,3}$$ . The only fusion products we will need can be obtained from [29] and (2.4): $$\Lambda_{2}$$ × $$\Lambda_{2}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{1}$$ + $$(2\Lambda_{2})_{2}$$ $$\Lambda_{2}$$ × $$\Lambda_{2}$$ × $$\Lambda_{2}=(0)_{1}$$ + $$2\,\pmb{\nabla}\,(\Lambda_{1})_{22}$$ + $$4\,\pmb{\mathrm{{E}}}\left(\Lambda_{2}\right)_{1122}$$ + $$3\,\pm\,(2\Lambda_{2})_{222}$$ + $$2\,\Xi\,(\Lambda_{1}+\Lambda_{2})_{33}$$ + $$(3\Lambda_{2})_{3}$$ Theorem 3.G2. The only nontrivial fusion-symmetries for $${G_{2}^{(1)}}$$ are $$(\pi_{3})^{\pm1}$$ at $$k=3$$ , and $$\pi_{4}$$ at $$k=4$$ . # 4. The Arguments The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty. Lemma 2.2 implies that a fusion-symmetry $$\pi$$ preserves q-dimensions: $${\mathcal{D}}(\lambda)={\mathcal{D}}(\pi\lambda)$$ $$\forall\lambda\in P_{+}$$ . In this subsection we use that to find a weight $$\Lambda_{\star}$$ for each algebra which must be essentially fixed by $$\pi$$ .	<p>There are Galois fusion-symmetries at levels $$k=3$$ and 4. In particular, for $$k=3$$ we have the fusion-symmetry $$\pi_{3}=\pi\{5\}$$ which interchanges both $$\Lambda_{2}\leftrightarrow\Lambda_{4}$$ and $$\Lambda_{1}\leftrightarrow3\Lambda_{4}$$ , and fixes the other five weights in $$P_{+}$$ . The exceptional $$\pi_{3}$$ was found independently in [34,14]. For $$k\,=\,4$$ we get a fusion-symmetry of order 4, which we will call $$\pi_{4}$$ . It fixes 0, $$\Lambda_{2}+\Lambda_{4}$$ , $$\Lambda_{3}+\Lambda_{4}$$ , and $$2\Lambda_{4}$$ , and permutes $$\Lambda_{4}\;\mapsto\;\Lambda_{1}\;\mapsto\;2\Lambda_{1}\;\mapsto\;4\Lambda_{4}\;\mapsto\;\Lambda_{4}$$ , $$\Lambda_{2}\mapsto2\Lambda_{3}\mapsto3\Lambda_{4}\mapsto\Lambda_{3}\mapsto\Lambda_{2}$$ , and $$\Lambda_{1}\!+\!\Lambda_{3}\mapsto\Lambda_{3}\!+\!2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{4}\mapsto\Lambda_{1}+2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{3}$$ . Its square $$\pi_{4}^{2}$$ equals the fusion-symmetry $$\pi\{5\}$$ .</p> <p>The only fusion products we need can be obtained from [29] and (2.4):</p> <p>$$\Lambda_{4}$$ × $$\Lambda_{4}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{1}$$ + $$(2\Lambda_{4})_{2}$$ $$\Lambda_{1}$$ × $$\Lambda_{4}=(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ $$\Lambda_{3}$$ × $$\Lambda_{4}=(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{3}$$ + $$(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ + $$(\Lambda_{3}+\Lambda_{4})_{3}$$ + $$(2\Lambda_{4})_{2}$$ $$(2\Lambda_{4})$$ × $$\Lambda_{4}=(\Lambda_{3})_{2}$$ + $$(\Lambda_{4})_{2}$$ + $$(2\Lambda_{4})_{2}$$ + $$(3\Lambda_{4})_{3}$$ + $$(\Lambda_{1}+\Lambda_{4})_{3}$$ + $$(\Lambda_{3}+\Lambda_{4})_{3}$$</p> <p>Theorem 3.F4. The only nontrivial fusion-symmetries of $${F}_{4}^{(1)}$$ are $$\pi_{3}$$ at level 3, and $$\pi_{4}^{i}$$ for $$1\leq i\leq3$$ , which occur at level 4.</p> <h1>3.9. The algebra $${G_{2}^{(1)}}$$</h1> <p>A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+\lambda_{2}$$ , and $$\kappa=k+4$$ . The conjugations and simple-currents are all trivial.</p> <p>Again there are nontrivial Galois fusion-symmetries. At $$k=3$$ , we have the order 3 fusion-symmetry $$\pi_{3}=\pi\{4\}$$ sending $$\Lambda_{1}\mapsto3\Lambda_{2}\mapsto\Lambda_{2}\mapsto\Lambda_{1}$$ , and fixing the remaining three weights. It was found in [14]. At $$k=4$$ , we have $$\pi_{4}=\pi\{5\}$$ permuting both $$\Lambda_{1}\leftrightarrow4\Lambda_{2}$$ and $$2\Lambda_{1}\leftrightarrow\Lambda_{2}$$ , and fixing the other five weights. It was found independently in [34,14], and in $$\S5$$ we will see that it is closely related to the $$\pi_{3}$$ of $$F_{4,3}$$ .</p> <p>The only fusion products we will need can be obtained from [29] and (2.4):</p> <p>$$\Lambda_{2}$$ × $$\Lambda_{2}=(0)_{1}$$ + $$(\Lambda_{1})_{2}$$ + $$(\Lambda_{2})_{1}$$ + $$(2\Lambda_{2})_{2}$$ $$\Lambda_{2}$$ × $$\Lambda_{2}$$ × $$\Lambda_{2}=(0)_{1}$$ + $$2\,\pmb{\nabla}\,(\Lambda_{1})_{22}$$ + $$4\,\pmb{\mathrm{{E}}}\left(\Lambda_{2}\right)_{1122}$$ + $$3\,\pm\,(2\Lambda_{2})_{222}$$ + $$2\,\Xi\,(\Lambda_{1}+\Lambda_{2})_{33}$$ + $$(3\Lambda_{2})_{3}$$</p> <p>Theorem 3.G2. The only nontrivial fusion-symmetries for $${G_{2}^{(1)}}$$ are $$(\pi_{3})^{\pm1}$$ at $$k=3$$ , and $$\pi_{4}$$ at $$k=4$$ .</p> <h1>4. The Arguments</h1> <p>The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty.</p> <p>Lemma 2.2 implies that a fusion-symmetry $$\pi$$ preserves q-dimensions: $${\mathcal{D}}(\lambda)={\mathcal{D}}(\pi\lambda)$$ $$\forall\lambda\in P_{+}$$ . In this subsection we use that to find a weight $$\Lambda_{\star}$$ for each algebra which must be essentially fixed by $$\pi$$ .</p>	[{"type": "text", "coordinates": [70, 70, 541, 172], "content": "There are Galois fusion-symmetries at levels $$k=3$$ and 4. In particular, for $$k=3$$ we\nhave the fusion-symmetry $$\\pi_{3}=\\pi\\{5\\}$$ which interchanges both $$\\Lambda_{2}\\leftrightarrow\\Lambda_{4}$$ and $$\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}$$ ,\nand fixes the other five weights in $$P_{+}$$ . The exceptional $$\\pi_{3}$$ was found independently\nin [34,14]. For $$k\\,=\\,4$$ we get a fusion-symmetry of order 4, which we will call $$\\pi_{4}$$ . It\nfixes 0, $$\\Lambda_{2}+\\Lambda_{4}$$ , $$\\Lambda_{3}+\\Lambda_{4}$$ , and $$2\\Lambda_{4}$$ , and permutes $$\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}$$ ,\n$$\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}$$ , and $$\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}$$ .\nIts square $$\\pi_{4}^{2}$$ equals the fusion-symmetry $$\\pi\\{5\\}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [95, 172, 467, 186], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [86, 197, 531, 270], "content": "$$\\Lambda_{4}$$ \u00d7 $$\\Lambda_{4}=(0)_{1}$$ + $$(\\Lambda_{1})_{2}$$ + $$(\\Lambda_{3})_{2}$$ + $$(\\Lambda_{4})_{1}$$ + $$(2\\Lambda_{4})_{2}$$\n$$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{4}=(\\Lambda_{3})_{2}$$ + $$(\\Lambda_{4})_{2}$$ + $$(\\Lambda_{1}+\\Lambda_{4})_{3}$$\n$$\\Lambda_{3}$$ \u00d7 $$\\Lambda_{4}=(\\Lambda_{1})_{2}$$ + $$(\\Lambda_{2})_{3}$$ + $$(\\Lambda_{3})_{2}$$ + $$(\\Lambda_{4})_{2}$$ + $$(\\Lambda_{1}+\\Lambda_{4})_{3}$$ + $$(\\Lambda_{3}+\\Lambda_{4})_{3}$$ + $$(2\\Lambda_{4})_{2}$$\n$$(2\\Lambda_{4})$$ \u00d7 $$\\Lambda_{4}=(\\Lambda_{3})_{2}$$ + $$(\\Lambda_{4})_{2}$$ + $$(2\\Lambda_{4})_{2}$$ + $$(3\\Lambda_{4})_{3}$$ + $$(\\Lambda_{1}+\\Lambda_{4})_{3}$$ + $$(\\Lambda_{3}+\\Lambda_{4})_{3}$$", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [71, 282, 542, 312], "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of $${F}_{4}^{(1)}$$ are $$\\pi_{3}$$ at level 3, and\n$$\\pi_{4}^{i}$$ for $$1\\leq i\\leq3$$ , which occur at level 4.", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [71, 326, 183, 343], "content": "3.9. The algebra $${G_{2}^{(1)}}$$", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [70, 350, 541, 378], "content": "A weight $$\\lambda$$ in $$P_{+}$$ satisfies $$k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}$$ , and $$\\kappa=k+4$$ . The conjugations and\nsimple-currents are all trivial.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 379, 541, 451], "content": "Again there are nontrivial Galois fusion-symmetries. At $$k=3$$ , we have the order 3\nfusion-symmetry $$\\pi_{3}=\\pi\\{4\\}$$ sending $$\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}$$ , and fixing the remaining three\nweights. It was found in [14]. At $$k=4$$ , we have $$\\pi_{4}=\\pi\\{5\\}$$ permuting both $$\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}$$\nand $$2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}$$ , and fixing the other five weights. It was found independently in [34,14],\nand in $$\\S5$$ we will see that it is closely related to the $$\\pi_{3}$$ of $$F_{4,3}$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [95, 451, 490, 466], "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [66, 477, 545, 513], "content": "$$\\Lambda_{2}$$ \u00d7 $$\\Lambda_{2}=(0)_{1}$$ + $$(\\Lambda_{1})_{2}$$ + $$(\\Lambda_{2})_{1}$$ + $$(2\\Lambda_{2})_{2}$$\n$$\\Lambda_{2}$$ \u00d7 $$\\Lambda_{2}$$ \u00d7 $$\\Lambda_{2}=(0)_{1}$$ + $$2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}$$ + $$4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}$$ + $$3\\,\\pm\\,(2\\Lambda_{2})_{222}$$ + $$2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}$$ + $$(3\\Lambda_{2})_{3}$$", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 525, 542, 556], "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for $${G_{2}^{(1)}}$$ are $$(\\pi_{3})^{\\pm1}$$ at $$k=3$$ ,\nand $$\\pi_{4}$$ at $$k=4$$ .", "block_type": "text", "index": 10}, {"type": "title", "coordinates": [249, 583, 362, 598], "content": "4. The Arguments", "block_type": "title", "index": 11}, {"type": "text", "coordinates": [70, 612, 541, 669], "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras\nis so accessible is (2.1b), which reduces the problem to studying Lie group characters at\nelements of finite order. These values have been studied by a number of people \u2014 see e.g.\n[22,28] \u2014 and the resulting combinatorics is often quite pretty.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [69, 670, 540, 713], "content": "Lemma 2.2 implies that a fusion-symmetry $$\\pi$$ preserves q-dimensions: $${\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)$$\n$$\\forall\\lambda\\in P_{+}$$ . In this subsection we use that to find a weight $$\\Lambda_{\\star}$$ for each algebra which must\nbe essentially fixed by $$\\pi$$ .", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [94, 73, 330, 88], "content": "There are Galois fusion-symmetries at levels ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [331, 75, 360, 84], "content": "k=3", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [360, 73, 492, 88], "content": " and 4. In particular, for ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [493, 75, 522, 84], "content": "k=3", "score": 0.91, "index": 4}, {"type": "text", "coordinates": [522, 73, 541, 88], "content": " we", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 87, 212, 104], "content": "have the fusion-symmetry ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [212, 89, 268, 101], "content": "\\pi_{3}=\\pi\\{5\\}", "score": 0.94, "index": 7}, {"type": "text", "coordinates": [268, 87, 405, 104], "content": " which interchanges both ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [405, 90, 453, 100], "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [454, 87, 481, 104], "content": " and ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [482, 90, 536, 101], "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [537, 87, 540, 104], "content": ",", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [70, 101, 265, 118], "content": "and fixes the other five weights in ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [265, 104, 281, 116], "content": "P_{+}", "score": 0.92, "index": 14}, {"type": "text", "coordinates": [282, 101, 387, 118], "content": ". The exceptional ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [387, 108, 399, 115], "content": "\\pi_{3}", "score": 0.88, "index": 16}, {"type": "text", "coordinates": [399, 101, 540, 118], "content": " was found independently", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [69, 116, 156, 132], "content": "in [34,14]. For ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [156, 118, 191, 128], "content": "k\\,=\\,4", "score": 0.87, "index": 19}, {"type": "text", "coordinates": [191, 116, 505, 132], "content": " we get a fusion-symmetry of order 4, which we will call ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [505, 122, 517, 129], "content": "\\pi_{4}", "score": 0.79, "index": 21}, {"type": "text", "coordinates": [518, 116, 542, 132], "content": ". It", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [69, 130, 114, 146], "content": "fixes 0, ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [115, 131, 160, 144], "content": "\\Lambda_{2}+\\Lambda_{4}", "score": 0.87, "index": 24}, {"type": "text", "coordinates": [161, 130, 168, 146], "content": ", ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [169, 131, 214, 144], "content": "\\Lambda_{3}+\\Lambda_{4}", "score": 0.87, "index": 26}, {"type": "text", "coordinates": [215, 130, 248, 146], "content": ", and ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [248, 131, 269, 144], "content": "2\\Lambda_{4}", "score": 0.88, "index": 28}, {"type": "text", "coordinates": [269, 130, 356, 146], "content": ", and permutes ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [356, 132, 536, 144], "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "score": 0.92, "index": 30}, {"type": "text", "coordinates": [537, 130, 541, 146], "content": ",", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [71, 145, 226, 158], "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [227, 145, 256, 160], "content": ", and ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [256, 146, 536, 158], "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "score": 0.81, "index": 34}, {"type": "text", "coordinates": [537, 145, 540, 160], "content": ".", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 159, 126, 174], "content": "Its square ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [126, 159, 140, 173], "content": "\\pi_{4}^{2}", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [140, 159, 290, 174], "content": " equals the fusion-symmetry ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [290, 159, 317, 173], "content": "\\pi\\{5\\}", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [317, 159, 321, 174], "content": ".", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [95, 173, 466, 189], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [97, 201, 112, 215], "content": "\\Lambda_{4}", "score": 0.85, "index": 42}, {"type": "text", "coordinates": [113, 200, 128, 217], "content": " \u00d7", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [128, 200, 179, 216], "content": "\\Lambda_{4}=(0)_{1}", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [180, 200, 196, 217], "content": " + ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [196, 201, 226, 216], "content": "(\\Lambda_{1})_{2}", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [226, 200, 243, 217], "content": " + ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [243, 200, 272, 216], "content": "(\\Lambda_{3})_{2}", "score": 0.9, "index": 48}, {"type": "text", "coordinates": [273, 200, 289, 217], "content": " + ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [290, 200, 318, 216], "content": "(\\Lambda_{4})_{1}", "score": 0.85, "index": 50}, {"type": "text", "coordinates": [319, 200, 336, 217], "content": " + ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [336, 201, 371, 216], "content": "(2\\Lambda_{4})_{2}", "score": 0.8, "index": 52}, {"type": "inline_equation", "coordinates": [97, 219, 112, 233], "content": "\\Lambda_{1}", "score": 0.88, "index": 53}, {"type": "text", "coordinates": [112, 220, 128, 235], "content": " \u00d7", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [129, 219, 187, 234], "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [187, 220, 204, 235], "content": " + ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [204, 219, 234, 234], "content": "(\\Lambda_{4})_{2}", "score": 0.9, "index": 57}, {"type": "text", "coordinates": [234, 220, 250, 235], "content": " + ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [250, 219, 309, 234], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "score": 0.88, "index": 59}, {"type": "inline_equation", "coordinates": [97, 237, 112, 251], "content": "\\Lambda_{3}", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [113, 237, 128, 253], "content": " \u00d7", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [128, 236, 187, 252], "content": "\\Lambda_{4}=(\\Lambda_{1})_{2}", "score": 0.93, "index": 62}, {"type": "text", "coordinates": [187, 237, 204, 253], "content": " + ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [204, 236, 234, 252], "content": "(\\Lambda_{2})_{3}", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [234, 237, 250, 253], "content": " + ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [251, 236, 280, 252], "content": "(\\Lambda_{3})_{2}", "score": 0.92, "index": 66}, {"type": "text", "coordinates": [280, 237, 297, 253], "content": " + ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [297, 236, 326, 252], "content": "(\\Lambda_{4})_{2}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [327, 237, 343, 253], "content": " + ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [344, 236, 402, 252], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [402, 237, 418, 253], "content": " + ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [419, 236, 476, 252], "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [477, 237, 493, 253], "content": " + ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [493, 236, 529, 252], "content": "(2\\Lambda_{4})_{2}", "score": 0.88, "index": 74}, {"type": "inline_equation", "coordinates": [82, 254, 111, 270], "content": "(2\\Lambda_{4})", "score": 0.88, "index": 75}, {"type": "text", "coordinates": [112, 254, 128, 271], "content": " \u00d7", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [128, 254, 187, 270], "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "score": 0.93, "index": 77}, {"type": "text", "coordinates": [187, 254, 204, 271], "content": " + ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [204, 254, 234, 270], "content": "(\\Lambda_{4})_{2}", "score": 0.91, "index": 79}, {"type": "text", "coordinates": [234, 254, 251, 271], "content": " + ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [251, 254, 286, 270], "content": "(2\\Lambda_{4})_{2}", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [286, 254, 303, 271], "content": " + ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [304, 254, 339, 270], "content": "(3\\Lambda_{4})_{3}", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [339, 254, 355, 271], "content": " + ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [356, 254, 414, 270], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "score": 0.9, "index": 85}, {"type": "text", "coordinates": [414, 254, 430, 271], "content": " + ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [431, 254, 488, 270], "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "score": 0.85, "index": 87}, {"type": "text", "coordinates": [90, 281, 403, 306], "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [403, 282, 427, 300], "content": "{F}_{4}^{(1)}", "score": 0.92, "index": 89}, {"type": "text", "coordinates": [427, 281, 449, 306], "content": "are ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [449, 287, 463, 299], "content": "\\pi_{3}", "score": 0.83, "index": 91}, {"type": "text", "coordinates": [464, 281, 542, 306], "content": " at level 3, and", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [70, 299, 84, 314], "content": "\\pi_{4}^{i}", "score": 0.9, "index": 93}, {"type": "text", "coordinates": [84, 300, 105, 315], "content": " for", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [106, 300, 156, 313], "content": "1\\leq i\\leq3", "score": 0.88, "index": 95}, {"type": "text", "coordinates": [156, 300, 281, 315], "content": ", which occur at level 4.", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [67, 328, 160, 348], "content": "3.9. The algebra ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [161, 327, 183, 345], "content": "{G_{2}^{(1)}}", "score": 0.91, "index": 98}, {"type": "text", "coordinates": [93, 352, 146, 368], "content": "A weight ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [146, 353, 154, 363], "content": "\\lambda", "score": 0.8, "index": 100}, {"type": "text", "coordinates": [155, 352, 172, 368], "content": " in ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [172, 353, 188, 366], "content": "P_{+}", "score": 0.9, "index": 102}, {"type": "text", "coordinates": [188, 352, 235, 368], "content": " satisfies ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [236, 352, 333, 365], "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "score": 0.92, "index": 104}, {"type": "text", "coordinates": [334, 352, 364, 368], "content": ", and ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [364, 354, 416, 364], "content": "\\kappa=k+4", "score": 0.9, "index": 106}, {"type": "text", "coordinates": [416, 352, 542, 368], "content": ". The conjugations and", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [71, 367, 228, 381], "content": "simple-currents are all trivial.", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [95, 381, 396, 396], "content": "Again there are nontrivial Galois fusion-symmetries. At ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [396, 383, 427, 392], "content": "k=3", "score": 0.89, "index": 110}, {"type": "text", "coordinates": [427, 381, 541, 396], "content": ", we have the order 3", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [69, 394, 160, 411], "content": "fusion-symmetry", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [161, 396, 215, 409], "content": "\\pi_{3}=\\pi\\{4\\}", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [216, 394, 259, 411], "content": " sending", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [260, 397, 377, 408], "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [378, 394, 541, 411], "content": ", and fixing the remaining three", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [71, 409, 252, 425], "content": "weights. It was found in [14]. At ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [252, 410, 284, 421], "content": "k=4", "score": 0.87, "index": 118}, {"type": "text", "coordinates": [285, 409, 338, 425], "content": ", we have ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [338, 411, 394, 423], "content": "\\pi_{4}=\\pi\\{5\\}", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [394, 409, 485, 425], "content": " permuting both ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [485, 412, 540, 423], "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "score": 0.91, "index": 122}, {"type": "text", "coordinates": [70, 423, 94, 439], "content": "and ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [94, 424, 149, 437], "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "score": 0.92, "index": 124}, {"type": "text", "coordinates": [150, 423, 541, 439], "content": ", and fixing the other five weights. It was found independently in [34,14],", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [70, 437, 108, 455], "content": "and in ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [108, 438, 120, 452], "content": "\\S5", "score": 0.39, "index": 127}, {"type": "text", "coordinates": [121, 437, 345, 455], "content": " we will see that it is closely related to the", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [345, 440, 359, 451], "content": "\\pi_{3}", "score": 0.84, "index": 129}, {"type": "text", "coordinates": [360, 437, 375, 455], "content": " of", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [376, 438, 397, 453], "content": "F_{4,3}", "score": 0.91, "index": 131}, {"type": "text", "coordinates": [398, 437, 401, 455], "content": ".", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [95, 452, 489, 468], "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [98, 479, 114, 493], "content": "\\Lambda_{2}", "score": 0.87, "index": 134}, {"type": "text", "coordinates": [114, 479, 130, 497], "content": " \u00d7 ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [130, 479, 181, 495], "content": "\\Lambda_{2}=(0)_{1}", "score": 0.93, "index": 136}, {"type": "text", "coordinates": [181, 479, 198, 497], "content": " + ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [198, 479, 227, 495], "content": "(\\Lambda_{1})_{2}", "score": 0.91, "index": 138}, {"type": "text", "coordinates": [228, 479, 244, 497], "content": " + ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [244, 479, 273, 495], "content": "(\\Lambda_{2})_{1}", "score": 0.91, "index": 140}, {"type": "text", "coordinates": [274, 479, 291, 497], "content": " + ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [291, 479, 326, 495], "content": "(2\\Lambda_{2})_{2}", "score": 0.89, "index": 142}, {"type": "inline_equation", "coordinates": [66, 497, 82, 511], "content": "\\Lambda_{2}", "score": 0.88, "index": 143}, {"type": "text", "coordinates": [82, 497, 97, 516], "content": " \u00d7", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [98, 498, 114, 511], "content": "\\Lambda_{2}", "score": 0.9, "index": 145}, {"type": "text", "coordinates": [114, 497, 129, 516], "content": " \u00d7", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [130, 497, 180, 513], "content": "\\Lambda_{2}=(0)_{1}", "score": 0.94, "index": 147}, {"type": "text", "coordinates": [181, 497, 197, 516], "content": " + ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [197, 497, 246, 513], "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "score": 0.91, "index": 149}, {"type": "text", "coordinates": [246, 497, 262, 516], "content": " +", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [263, 497, 321, 513], "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "score": 0.87, "index": 151}, {"type": "text", "coordinates": [322, 497, 338, 516], "content": " + ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [338, 497, 398, 513], "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "score": 0.89, "index": 153}, {"type": "text", "coordinates": [398, 497, 414, 516], "content": " + ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [414, 497, 491, 513], "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "score": 0.89, "index": 155}, {"type": "text", "coordinates": [492, 497, 509, 516], "content": " + ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [509, 497, 544, 513], "content": "(3\\Lambda_{2})_{3}", "score": 0.89, "index": 157}, {"type": "text", "coordinates": [91, 522, 408, 548], "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [409, 525, 432, 543], "content": "{G_{2}^{(1)}}", "score": 0.92, "index": 159}, {"type": "text", "coordinates": [433, 522, 455, 548], "content": "are ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [455, 527, 490, 543], "content": "(\\pi_{3})^{\\pm1}", "score": 0.91, "index": 161}, {"type": "text", "coordinates": [491, 522, 506, 548], "content": " at ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [506, 528, 537, 541], "content": "k=3", "score": 0.87, "index": 163}, {"type": "text", "coordinates": [537, 522, 543, 548], "content": ",", "score": 1.0, "index": 164}, {"type": "text", "coordinates": [72, 544, 93, 557], "content": "and", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [93, 544, 107, 556], "content": "\\pi_{4}", "score": 0.84, "index": 166}, {"type": "text", "coordinates": [108, 544, 124, 557], "content": " at ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [124, 543, 155, 555], "content": "k=4", "score": 0.89, "index": 168}, {"type": "text", "coordinates": [155, 544, 160, 557], "content": ".", "score": 1.0, "index": 169}, {"type": "text", "coordinates": [249, 585, 362, 599], "content": "4. The Arguments", "score": 1.0, "index": 170}, {"type": "text", "coordinates": [95, 614, 540, 628], "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "score": 1.0, "index": 171}, {"type": "text", "coordinates": [69, 628, 541, 642], "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "score": 1.0, "index": 172}, {"type": "text", "coordinates": [70, 641, 541, 659], "content": "elements of finite order. These values have been studied by a number of people \u2014 see e.g.", "score": 1.0, "index": 173}, {"type": "text", "coordinates": [71, 656, 401, 673], "content": "[22,28] \u2014 and the resulting combinatorics is often quite pretty.", "score": 1.0, "index": 174}, {"type": "text", "coordinates": [93, 671, 325, 686], "content": "Lemma 2.2 implies that a fusion-symmetry ", "score": 1.0, "index": 175}, {"type": "inline_equation", "coordinates": [325, 677, 332, 682], "content": "\\pi", "score": 0.87, "index": 176}, {"type": "text", "coordinates": [333, 671, 464, 686], "content": " preserves q-dimensions: ", "score": 1.0, "index": 177}, {"type": "inline_equation", "coordinates": [464, 673, 540, 685], "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "score": 0.94, "index": 178}, {"type": "inline_equation", "coordinates": [70, 688, 116, 699], "content": "\\forall\\lambda\\in P_{+}", "score": 0.92, "index": 179}, {"type": "text", "coordinates": [116, 686, 373, 701], "content": ". In this subsection we use that to find a weight ", "score": 1.0, "index": 180}, {"type": "inline_equation", "coordinates": [374, 688, 387, 698], "content": "\\Lambda_{\\star}", "score": 0.92, "index": 181}, {"type": "text", "coordinates": [388, 686, 540, 701], "content": "for each algebra which must", "score": 1.0, "index": 182}, {"type": "text", "coordinates": [70, 700, 190, 715], "content": "be essentially fixed by ", "score": 1.0, "index": 183}, {"type": "inline_equation", "coordinates": [190, 705, 198, 711], "content": "\\pi", "score": 0.88, "index": 184}, {"type": "text", "coordinates": [198, 700, 203, 715], "content": ".", "score": 1.0, "index": 185}]	[]	[{"type": "inline", "coordinates": [331, 75, 360, 84], "content": "k=3", "caption": ""}, {"type": "inline", "coordinates": [493, 75, 522, 84], "content": "k=3", "caption": ""}, {"type": "inline", "coordinates": [212, 89, 268, 101], "content": "\\pi_{3}=\\pi\\{5\\}", "caption": ""}, {"type": "inline", "coordinates": [405, 90, 453, 100], "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [482, 90, 536, 101], "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [265, 104, 281, 116], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [387, 108, 399, 115], "content": "\\pi_{3}", "caption": ""}, {"type": "inline", "coordinates": [156, 118, 191, 128], "content": "k\\,=\\,4", "caption": ""}, {"type": "inline", "coordinates": [505, 122, 517, 129], "content": "\\pi_{4}", "caption": ""}, {"type": "inline", "coordinates": [115, 131, 160, 144], "content": "\\Lambda_{2}+\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [169, 131, 214, 144], "content": "\\Lambda_{3}+\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [248, 131, 269, 144], "content": "2\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [356, 132, 536, 144], "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [71, 145, 226, 158], "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [256, 146, 536, 158], "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "caption": ""}, {"type": "inline", "coordinates": [126, 159, 140, 173], "content": "\\pi_{4}^{2}", "caption": ""}, {"type": "inline", "coordinates": [290, 159, 317, 173], "content": "\\pi\\{5\\}", "caption": ""}, {"type": "inline", "coordinates": [97, 201, 112, 215], "content": "\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [128, 200, 179, 216], "content": "\\Lambda_{4}=(0)_{1}", "caption": ""}, {"type": 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"inline", "coordinates": [146, 353, 154, 363], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [172, 353, 188, 366], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [236, 352, 333, 365], "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [364, 354, 416, 364], "content": "\\kappa=k+4", "caption": ""}, {"type": "inline", "coordinates": [396, 383, 427, 392], "content": "k=3", "caption": ""}, {"type": "inline", "coordinates": [161, 396, 215, 409], "content": "\\pi_{3}=\\pi\\{4\\}", "caption": ""}, {"type": "inline", "coordinates": [260, 397, 377, 408], "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [252, 410, 284, 421], "content": "k=4", "caption": ""}, {"type": "inline", "coordinates": [338, 411, 394, 423], "content": "\\pi_{4}=\\pi\\{5\\}", "caption": ""}, {"type": "inline", "coordinates": [485, 412, 540, 423], "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [94, 424, 149, 437], "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [108, 438, 120, 452], "content": "\\S5", "caption": ""}, {"type": "inline", "coordinates": [345, 440, 359, 451], "content": "\\pi_{3}", "caption": ""}, {"type": "inline", "coordinates": [376, 438, 397, 453], "content": "F_{4,3}", "caption": ""}, {"type": "inline", "coordinates": [98, 479, 114, 493], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [130, 479, 181, 495], "content": "\\Lambda_{2}=(0)_{1}", "caption": ""}, {"type": "inline", "coordinates": [198, 479, 227, 495], "content": "(\\Lambda_{1})_{2}", "caption": ""}, {"type": "inline", "coordinates": [244, 479, 273, 495], "content": "(\\Lambda_{2})_{1}", "caption": ""}, {"type": "inline", "coordinates": [291, 479, 326, 495], "content": "(2\\Lambda_{2})_{2}", "caption": ""}, {"type": "inline", "coordinates": [66, 497, 82, 511], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [98, 498, 114, 511], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [130, 497, 180, 513], "content": "\\Lambda_{2}=(0)_{1}", "caption": ""}, {"type": "inline", "coordinates": [197, 497, 246, 513], "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "caption": ""}, {"type": "inline", "coordinates": [263, 497, 321, 513], "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "caption": ""}, {"type": "inline", "coordinates": [338, 497, 398, 513], "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "caption": ""}, {"type": "inline", "coordinates": [414, 497, 491, 513], "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "caption": ""}, {"type": "inline", "coordinates": [509, 497, 544, 513], "content": "(3\\Lambda_{2})_{3}", "caption": ""}, {"type": "inline", "coordinates": [409, 525, 432, 543], 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"text": "There are Galois fusion-symmetries at levels $k=3$ and 4. In particular, for $k=3$ we have the fusion-symmetry $\\pi_{3}=\\pi\\{5\\}$ which interchanges both $\\Lambda_{2}\\leftrightarrow\\Lambda_{4}$ and $\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}$ , and fixes the other five weights in $P_{+}$ . The exceptional $\\pi_{3}$ was found independently in [34,14]. For $k\\,=\\,4$ we get a fusion-symmetry of order 4, which we will call $\\pi_{4}$ . It fixes 0, $\\Lambda_{2}+\\Lambda_{4}$ , $\\Lambda_{3}+\\Lambda_{4}$ , and $2\\Lambda_{4}$ , and permutes $\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}$ , $\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}$ , and $\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}$ . Its square $\\pi_{4}^{2}$ equals the fusion-symmetry $\\pi\\{5\\}$ . ", "page_idx": 14}, {"type": "text", "text": "The only fusion products we need can be obtained from [29] and (2.4): ", "page_idx": 14}, {"type": "text", "text": "$\\Lambda_{4}$ \u00d7 $\\Lambda_{4}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{1}$ + $(2\\Lambda_{4})_{2}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ $\\Lambda_{3}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ + $(2\\Lambda_{4})_{2}$ $(2\\Lambda_{4})$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(2\\Lambda_{4})_{2}$ + $(3\\Lambda_{4})_{3}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ ", "page_idx": 14}, {"type": "text", "text": "Theorem 3.F4. The only nontrivial fusion-symmetries of ${F}_{4}^{(1)}$ are $\\pi_{3}$ at level 3, and $\\pi_{4}^{i}$ for $1\\leq i\\leq3$ , which occur at level 4. ", "page_idx": 14}, {"type": "text", "text": "3.9. The algebra ${G_{2}^{(1)}}$ ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}$ , and $\\kappa=k+4$ . The conjugations and simple-currents are all trivial. ", "page_idx": 14}, {"type": "text", "text": "Again there are nontrivial Galois fusion-symmetries. At $k=3$ , we have the order 3 fusion-symmetry $\\pi_{3}=\\pi\\{4\\}$ sending $\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}$ , and fixing the remaining three weights. It was found in [14]. At $k=4$ , we have $\\pi_{4}=\\pi\\{5\\}$ permuting both $\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}$ and $2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}$ , and fixing the other five weights. It was found independently in [34,14], and in $\\S5$ we will see that it is closely related to the $\\pi_{3}$ of $F_{4,3}$ . ", "page_idx": 14}, {"type": "text", "text": "The only fusion products we will need can be obtained from [29] and (2.4): ", "page_idx": 14}, {"type": "text", "text": "$\\Lambda_{2}$ \u00d7 $\\Lambda_{2}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{1}$ + $(2\\Lambda_{2})_{2}$ $\\Lambda_{2}$ \u00d7 $\\Lambda_{2}$ \u00d7 $\\Lambda_{2}=(0)_{1}$ + $2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}$ + $4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}$ + $3\\,\\pm\\,(2\\Lambda_{2})_{222}$ + $2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}$ + $(3\\Lambda_{2})_{3}$ ", "page_idx": 14}, {"type": "text", "text": "Theorem 3.G2. The only nontrivial fusion-symmetries for ${G_{2}^{(1)}}$ are $(\\pi_{3})^{\\pm1}$ at $k=3$ , and $\\pi_{4}$ at $k=4$ . ", "page_idx": 14}, {"type": "text", "text": "4. The Arguments ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people \u2014 see e.g. [22,28] \u2014 and the resulting combinatorics is often quite pretty. ", "page_idx": 14}, {"type": "text", "text": "Lemma 2.2 implies that a fusion-symmetry $\\pi$ preserves q-dimensions: ${\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)$ $\\forall\\lambda\\in P_{+}$ . In this subsection we use that to find a weight $\\Lambda_{\\star}$ for each algebra which must be essentially fixed by $\\pi$ . 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0.85, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 200, 128, 217], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [128, 200, 179, 216], "score": 0.92, "content": "\\Lambda_{4}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [180, 200, 196, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [196, 201, 226, 216], "score": 0.9, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [226, 200, 243, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [243, 200, 272, 216], "score": 0.9, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [273, 200, 289, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [290, 200, 318, 216], "score": 0.85, "content": "(\\Lambda_{4})_{1}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [319, 200, 336, 217], "score": 1.0, 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\u00d7", "type": "text"}, {"bbox": [128, 254, 187, 270], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 254, 204, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 254, 234, 270], "score": 0.91, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 254, 251, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 254, 286, 270], "score": 0.91, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [286, 254, 303, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [304, 254, 339, 270], "score": 0.91, "content": "(3\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [339, 254, 355, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [356, 254, 414, 270], "score": 0.9, "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [414, 254, 430, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [431, 254, 488, 270], "score": 0.85, "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 57}], "index": 11}], "index": 9.5}, {"type": "text", "bbox": [71, 282, 542, 312], "lines": [{"bbox": [90, 281, 542, 306], "spans": [{"bbox": [90, 281, 403, 306], "score": 1.0, "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of ", "type": "text"}, {"bbox": [403, 282, 427, 300], "score": 0.92, "content": "{F}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [427, 281, 449, 306], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [449, 287, 463, 299], "score": 0.83, "content": "\\pi_{3}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [464, 281, 542, 306], "score": 1.0, "content": " at level 3, and", "type": "text"}], "index": 12}, {"bbox": [70, 299, 281, 315], "spans": [{"bbox": [70, 299, 84, 314], "score": 0.9, "content": "\\pi_{4}^{i}", "type": "inline_equation", "height": 15, "width": 14}, {"bbox": [84, 300, 105, 315], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [106, 300, 156, 313], "score": 0.88, "content": "1\\leq i\\leq3", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [156, 300, 281, 315], "score": 1.0, "content": ", which occur at level 4.", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "title", "bbox": [71, 326, 183, 343], "lines": [{"bbox": [67, 327, 183, 348], "spans": [{"bbox": [67, 328, 160, 348], "score": 1.0, "content": "3.9. The algebra ", "type": "text"}, {"bbox": [161, 327, 183, 345], "score": 0.91, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 22}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 350, 541, 378], "lines": [{"bbox": [93, 352, 542, 368], "spans": [{"bbox": [93, 352, 146, 368], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 353, 154, 363], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [155, 352, 172, 368], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [172, 353, 188, 366], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [188, 352, 235, 368], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [236, 352, 333, 365], "score": 0.92, "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [334, 352, 364, 368], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [364, 354, 416, 364], "score": 0.9, "content": "\\kappa=k+4", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [416, 352, 542, 368], "score": 1.0, "content": ". The conjugations and", "type": "text"}], "index": 15}, {"bbox": [71, 367, 228, 381], "spans": [{"bbox": [71, 367, 228, 381], "score": 1.0, "content": "simple-currents are all trivial.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [70, 379, 541, 451], "lines": [{"bbox": [95, 381, 541, 396], "spans": [{"bbox": [95, 381, 396, 396], "score": 1.0, "content": "Again there are nontrivial Galois fusion-symmetries. At ", "type": "text"}, {"bbox": [396, 383, 427, 392], "score": 0.89, "content": "k=3", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [427, 381, 541, 396], "score": 1.0, "content": ", we have the order 3", "type": "text"}], "index": 17}, {"bbox": [69, 394, 541, 411], "spans": [{"bbox": [69, 394, 160, 411], "score": 1.0, "content": "fusion-symmetry", "type": "text"}, {"bbox": [161, 396, 215, 409], "score": 0.92, "content": "\\pi_{3}=\\pi\\{4\\}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [216, 394, 259, 411], "score": 1.0, "content": " sending", "type": "text"}, {"bbox": [260, 397, 377, 408], "score": 0.92, "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 117}, {"bbox": [378, 394, 541, 411], "score": 1.0, "content": ", and fixing the remaining three", "type": "text"}], "index": 18}, {"bbox": [71, 409, 540, 425], "spans": [{"bbox": [71, 409, 252, 425], "score": 1.0, "content": "weights. It was found in [14]. At ", "type": "text"}, {"bbox": [252, 410, 284, 421], "score": 0.87, "content": "k=4", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 409, 338, 425], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [338, 411, 394, 423], "score": 0.93, "content": "\\pi_{4}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [394, 409, 485, 425], "score": 1.0, "content": " permuting both ", "type": "text"}, {"bbox": [485, 412, 540, 423], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 55}], "index": 19}, {"bbox": [70, 423, 541, 439], "spans": [{"bbox": [70, 423, 94, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 424, 149, 437], "score": 0.92, "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [150, 423, 541, 439], "score": 1.0, "content": ", and fixing the other five weights. It was found independently in [34,14],", "type": "text"}], "index": 20}, {"bbox": [70, 437, 401, 455], "spans": [{"bbox": [70, 437, 108, 455], "score": 1.0, "content": "and in ", "type": "text"}, {"bbox": [108, 438, 120, 452], "score": 0.39, "content": "\\S5", "type": "inline_equation", "height": 14, "width": 12}, {"bbox": [121, 437, 345, 455], "score": 1.0, "content": " we will see that it is closely related to the", "type": "text"}, {"bbox": [345, 440, 359, 451], "score": 0.84, "content": "\\pi_{3}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [360, 437, 375, 455], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [376, 438, 397, 453], "score": 0.91, "content": "F_{4,3}", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [398, 437, 401, 455], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [95, 451, 490, 466], "lines": [{"bbox": [95, 452, 489, 468], "spans": [{"bbox": [95, 452, 489, 468], "score": 1.0, "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [66, 477, 545, 513], "lines": [{"bbox": [98, 479, 326, 497], "spans": [{"bbox": [98, 479, 114, 493], "score": 0.87, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [114, 479, 130, 497], "score": 1.0, "content": " \u00d7 ", "type": "text"}, {"bbox": [130, 479, 181, 495], "score": 0.93, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [181, 479, 198, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [198, 479, 227, 495], "score": 0.91, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [228, 479, 244, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [244, 479, 273, 495], "score": 0.91, "content": "(\\Lambda_{2})_{1}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [274, 479, 291, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [291, 479, 326, 495], "score": 0.89, "content": "(2\\Lambda_{2})_{2}", "type": "inline_equation", "height": 16, "width": 35}], "index": 23}, {"bbox": [66, 497, 544, 516], "spans": [{"bbox": [66, 497, 82, 511], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [82, 497, 97, 516], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [98, 498, 114, 511], "score": 0.9, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [114, 497, 129, 516], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [130, 497, 180, 513], "score": 0.94, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 50}, {"bbox": [181, 497, 197, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [197, 497, 246, 513], "score": 0.91, "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "type": "inline_equation", "height": 16, "width": 49}, {"bbox": [246, 497, 262, 516], "score": 1.0, "content": " +", "type": "text"}, {"bbox": [263, 497, 321, 513], "score": 0.87, "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [322, 497, 338, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [338, 497, 398, 513], "score": 0.89, "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [398, 497, 414, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [414, 497, 491, 513], "score": 0.89, "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [492, 497, 509, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [509, 497, 544, 513], "score": 0.89, "content": "(3\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 35}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [70, 525, 542, 556], "lines": [{"bbox": [91, 522, 543, 548], "spans": [{"bbox": [91, 522, 408, 548], "score": 1.0, "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for", "type": "text"}, {"bbox": [409, 525, 432, 543], "score": 0.92, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [433, 522, 455, 548], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [455, 527, 490, 543], "score": 0.91, "content": "(\\pi_{3})^{\\pm1}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [491, 522, 506, 548], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [506, 528, 537, 541], "score": 0.87, "content": "k=3", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [537, 522, 543, 548], "score": 1.0, "content": ",", "type": "text"}], "index": 25}, {"bbox": [72, 543, 160, 557], "spans": [{"bbox": [72, 544, 93, 557], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [93, 544, 107, 556], "score": 0.84, "content": "\\pi_{4}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [108, 544, 124, 557], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [124, 543, 155, 555], "score": 0.89, "content": "k=4", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [155, 544, 160, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "title", "bbox": [249, 583, 362, 598], "lines": [{"bbox": [249, 585, 362, 599], "spans": [{"bbox": [249, 585, 362, 599], "score": 1.0, "content": "4. The Arguments", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [70, 612, 541, 669], "lines": [{"bbox": [95, 614, 540, 628], "spans": [{"bbox": [95, 614, 540, 628], "score": 1.0, "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "type": "text"}], "index": 28}, {"bbox": [69, 628, 541, 642], "spans": [{"bbox": [69, 628, 541, 642], "score": 1.0, "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "type": "text"}], "index": 29}, {"bbox": [70, 641, 541, 659], "spans": [{"bbox": [70, 641, 541, 659], "score": 1.0, "content": "elements of finite order. These values have been studied by a number of people \u2014 see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] \u2014 and the resulting combinatorics is often quite pretty.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [69, 670, 540, 713], "lines": [{"bbox": [93, 671, 540, 686], "spans": [{"bbox": [93, 671, 325, 686], "score": 1.0, "content": "Lemma 2.2 implies that a fusion-symmetry ", "type": "text"}, {"bbox": [325, 677, 332, 682], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [333, 671, 464, 686], "score": 1.0, "content": " preserves q-dimensions: ", "type": "text"}, {"bbox": [464, 673, 540, 685], "score": 0.94, "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "type": "inline_equation", "height": 12, "width": 76}], "index": 32}, {"bbox": [70, 686, 540, 701], "spans": [{"bbox": [70, 688, 116, 699], "score": 0.92, "content": "\\forall\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [116, 686, 373, 701], "score": 1.0, "content": ". In this subsection we use that to find a weight ", "type": "text"}, {"bbox": [374, 688, 387, 698], "score": 0.92, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [388, 686, 540, 701], "score": 1.0, "content": "for each algebra which must", "type": "text"}], "index": 33}, {"bbox": [70, 700, 203, 715], "spans": [{"bbox": [70, 700, 190, 715], "score": 1.0, "content": "be essentially fixed by ", "type": "text"}, {"bbox": [190, 705, 198, 711], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [198, 700, 203, 715], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "15", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 172], "lines": [{"bbox": [94, 73, 541, 88], "spans": [{"bbox": [94, 73, 330, 88], "score": 1.0, "content": "There are Galois fusion-symmetries at levels ", "type": "text"}, {"bbox": [331, 75, 360, 84], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [360, 73, 492, 88], "score": 1.0, "content": " and 4. In particular, for ", "type": "text"}, {"bbox": [493, 75, 522, 84], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [522, 73, 541, 88], "score": 1.0, "content": " we", "type": "text"}], "index": 0}, {"bbox": [70, 87, 540, 104], "spans": [{"bbox": [70, 87, 212, 104], "score": 1.0, "content": "have the fusion-symmetry ", "type": "text"}, {"bbox": [212, 89, 268, 101], "score": 0.94, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [268, 87, 405, 104], "score": 1.0, "content": " which interchanges both ", "type": "text"}, {"bbox": [405, 90, 453, 100], "score": 0.92, "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [454, 87, 481, 104], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [482, 90, 536, 101], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [537, 87, 540, 104], "score": 1.0, "content": ",", "type": "text"}], "index": 1}, {"bbox": [70, 101, 540, 118], "spans": [{"bbox": [70, 101, 265, 118], "score": 1.0, "content": "and fixes the other five weights in ", "type": "text"}, {"bbox": [265, 104, 281, 116], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [282, 101, 387, 118], "score": 1.0, "content": ". The exceptional ", "type": "text"}, {"bbox": [387, 108, 399, 115], "score": 0.88, "content": "\\pi_{3}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [399, 101, 540, 118], "score": 1.0, "content": " was found independently", "type": "text"}], "index": 2}, {"bbox": [69, 116, 542, 132], "spans": [{"bbox": [69, 116, 156, 132], "score": 1.0, "content": "in [34,14]. For ", "type": "text"}, {"bbox": [156, 118, 191, 128], "score": 0.87, "content": "k\\,=\\,4", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [191, 116, 505, 132], "score": 1.0, "content": " we get a fusion-symmetry of order 4, which we will call ", "type": "text"}, {"bbox": [505, 122, 517, 129], "score": 0.79, "content": "\\pi_{4}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [518, 116, 542, 132], "score": 1.0, "content": ". It", "type": "text"}], "index": 3}, {"bbox": [69, 130, 541, 146], "spans": [{"bbox": [69, 130, 114, 146], "score": 1.0, "content": "fixes 0, ", "type": "text"}, {"bbox": [115, 131, 160, 144], "score": 0.87, "content": "\\Lambda_{2}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [161, 130, 168, 146], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [169, 131, 214, 144], "score": 0.87, "content": "\\Lambda_{3}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [215, 130, 248, 146], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [248, 131, 269, 144], "score": 0.88, "content": "2\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [269, 130, 356, 146], "score": 1.0, "content": ", and permutes ", "type": "text"}, {"bbox": [356, 132, 536, 144], "score": 0.92, "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "type": "inline_equation", "height": 12, "width": 180}, {"bbox": [537, 130, 541, 146], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [71, 145, 540, 160], "spans": [{"bbox": [71, 145, 226, 158], "score": 0.92, "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 155}, {"bbox": [227, 145, 256, 160], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [256, 146, 536, 158], "score": 0.81, "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "type": "inline_equation", "height": 12, "width": 280}, {"bbox": [537, 145, 540, 160], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [70, 159, 321, 174], "spans": [{"bbox": [70, 159, 126, 174], "score": 1.0, "content": "Its square ", "type": "text"}, {"bbox": [126, 159, 140, 173], "score": 0.89, "content": "\\pi_{4}^{2}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [140, 159, 290, 174], "score": 1.0, "content": " equals the fusion-symmetry ", "type": "text"}, {"bbox": [290, 159, 317, 173], "score": 0.92, "content": "\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [317, 159, 321, 174], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 3, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [69, 73, 542, 174]}, {"type": "text", "bbox": [95, 172, 467, 186], "lines": [{"bbox": [95, 173, 466, 189], "spans": [{"bbox": [95, 173, 466, 189], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [95, 173, 466, 189]}, {"type": "text", "bbox": [86, 197, 531, 270], "lines": [{"bbox": [97, 200, 371, 217], "spans": [{"bbox": [97, 201, 112, 215], "score": 0.85, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 200, 128, 217], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [128, 200, 179, 216], "score": 0.92, "content": "\\Lambda_{4}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [180, 200, 196, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [196, 201, 226, 216], "score": 0.9, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [226, 200, 243, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [243, 200, 272, 216], "score": 0.9, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [273, 200, 289, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [290, 200, 318, 216], "score": 0.85, "content": "(\\Lambda_{4})_{1}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [319, 200, 336, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [336, 201, 371, 216], "score": 0.8, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 15, "width": 35}], "index": 8}, {"bbox": [97, 219, 309, 235], "spans": [{"bbox": [97, 219, 112, 233], "score": 0.88, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [112, 220, 128, 235], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [129, 219, 187, 234], "score": 0.92, "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [187, 220, 204, 235], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 219, 234, 234], "score": 0.9, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [234, 220, 250, 235], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [250, 219, 309, 234], "score": 0.88, "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 15, "width": 59}], "index": 9}, {"bbox": [97, 236, 529, 253], "spans": [{"bbox": [97, 237, 112, 251], "score": 0.88, "content": "\\Lambda_{3}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 237, 128, 253], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [128, 236, 187, 252], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 237, 204, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 236, 234, 252], "score": 0.92, "content": "(\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 237, 250, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 236, 280, 252], "score": 0.92, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [280, 237, 297, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [297, 236, 326, 252], "score": 0.92, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [327, 237, 343, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [344, 236, 402, 252], "score": 0.91, "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [402, 237, 418, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [419, 236, 476, 252], "score": 0.91, "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 57}, {"bbox": [477, 237, 493, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [493, 236, 529, 252], "score": 0.88, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 36}], "index": 10}, {"bbox": [82, 254, 488, 271], "spans": [{"bbox": [82, 254, 111, 270], "score": 0.88, "content": "(2\\Lambda_{4})", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [112, 254, 128, 271], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [128, 254, 187, 270], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 254, 204, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 254, 234, 270], "score": 0.91, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 254, 251, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 254, 286, 270], "score": 0.91, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [286, 254, 303, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [304, 254, 339, 270], "score": 0.91, "content": "(3\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [339, 254, 355, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [356, 254, 414, 270], "score": 0.9, "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [414, 254, 430, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [431, 254, 488, 270], "score": 0.85, "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 57}], "index": 11}], "index": 9.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [82, 200, 529, 271]}, {"type": "text", "bbox": [71, 282, 542, 312], "lines": [{"bbox": [90, 281, 542, 306], "spans": [{"bbox": [90, 281, 403, 306], "score": 1.0, "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of ", "type": "text"}, {"bbox": [403, 282, 427, 300], "score": 0.92, "content": "{F}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [427, 281, 449, 306], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [449, 287, 463, 299], "score": 0.83, "content": "\\pi_{3}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [464, 281, 542, 306], "score": 1.0, "content": " at level 3, and", "type": "text"}], "index": 12}, {"bbox": [70, 299, 281, 315], "spans": [{"bbox": [70, 299, 84, 314], "score": 0.9, "content": "\\pi_{4}^{i}", "type": "inline_equation", "height": 15, "width": 14}, {"bbox": [84, 300, 105, 315], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [106, 300, 156, 313], "score": 0.88, "content": "1\\leq i\\leq3", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [156, 300, 281, 315], "score": 1.0, "content": ", which occur at level 4.", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [70, 281, 542, 315]}, {"type": "title", "bbox": [71, 326, 183, 343], "lines": [{"bbox": [67, 327, 183, 348], "spans": [{"bbox": [67, 328, 160, 348], "score": 1.0, "content": "3.9. The algebra ", "type": "text"}, {"bbox": [161, 327, 183, 345], "score": 0.91, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 22}], "index": 14}], "index": 14, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 350, 541, 378], "lines": [{"bbox": [93, 352, 542, 368], "spans": [{"bbox": [93, 352, 146, 368], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 353, 154, 363], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [155, 352, 172, 368], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [172, 353, 188, 366], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [188, 352, 235, 368], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [236, 352, 333, 365], "score": 0.92, "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [334, 352, 364, 368], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [364, 354, 416, 364], "score": 0.9, "content": "\\kappa=k+4", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [416, 352, 542, 368], "score": 1.0, "content": ". The conjugations and", "type": "text"}], "index": 15}, {"bbox": [71, 367, 228, 381], "spans": [{"bbox": [71, 367, 228, 381], "score": 1.0, "content": "simple-currents are all trivial.", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [71, 352, 542, 381]}, {"type": "text", "bbox": [70, 379, 541, 451], "lines": [{"bbox": [95, 381, 541, 396], "spans": [{"bbox": [95, 381, 396, 396], "score": 1.0, "content": "Again there are nontrivial Galois fusion-symmetries. At ", "type": "text"}, {"bbox": [396, 383, 427, 392], "score": 0.89, "content": "k=3", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [427, 381, 541, 396], "score": 1.0, "content": ", we have the order 3", "type": "text"}], "index": 17}, {"bbox": [69, 394, 541, 411], "spans": [{"bbox": [69, 394, 160, 411], "score": 1.0, "content": "fusion-symmetry", "type": "text"}, {"bbox": [161, 396, 215, 409], "score": 0.92, "content": "\\pi_{3}=\\pi\\{4\\}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [216, 394, 259, 411], "score": 1.0, "content": " sending", "type": "text"}, {"bbox": [260, 397, 377, 408], "score": 0.92, "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 117}, {"bbox": [378, 394, 541, 411], "score": 1.0, "content": ", and fixing the remaining three", "type": "text"}], "index": 18}, {"bbox": [71, 409, 540, 425], "spans": [{"bbox": [71, 409, 252, 425], "score": 1.0, "content": "weights. It was found in [14]. At ", "type": "text"}, {"bbox": [252, 410, 284, 421], "score": 0.87, "content": "k=4", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 409, 338, 425], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [338, 411, 394, 423], "score": 0.93, "content": "\\pi_{4}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [394, 409, 485, 425], "score": 1.0, "content": " permuting both ", "type": "text"}, {"bbox": [485, 412, 540, 423], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 55}], "index": 19}, {"bbox": [70, 423, 541, 439], "spans": [{"bbox": [70, 423, 94, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 424, 149, 437], "score": 0.92, "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [150, 423, 541, 439], "score": 1.0, "content": ", and fixing the other five weights. It was found independently in [34,14],", "type": "text"}], "index": 20}, {"bbox": [70, 437, 401, 455], "spans": [{"bbox": [70, 437, 108, 455], "score": 1.0, "content": "and in ", "type": "text"}, {"bbox": [108, 438, 120, 452], "score": 0.39, "content": "\\S5", "type": "inline_equation", "height": 14, "width": 12}, {"bbox": [121, 437, 345, 455], "score": 1.0, "content": " we will see that it is closely related to the", "type": "text"}, {"bbox": [345, 440, 359, 451], "score": 0.84, "content": "\\pi_{3}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [360, 437, 375, 455], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [376, 438, 397, 453], "score": 0.91, "content": "F_{4,3}", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [398, 437, 401, 455], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [69, 381, 541, 455]}, {"type": "text", "bbox": [95, 451, 490, 466], "lines": [{"bbox": [95, 452, 489, 468], "spans": [{"bbox": [95, 452, 489, 468], "score": 1.0, "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [95, 452, 489, 468]}, {"type": "text", "bbox": [66, 477, 545, 513], "lines": [{"bbox": [98, 479, 326, 497], "spans": [{"bbox": [98, 479, 114, 493], "score": 0.87, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [114, 479, 130, 497], "score": 1.0, "content": " \u00d7 ", "type": "text"}, {"bbox": [130, 479, 181, 495], "score": 0.93, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [181, 479, 198, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [198, 479, 227, 495], "score": 0.91, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [228, 479, 244, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [244, 479, 273, 495], "score": 0.91, "content": "(\\Lambda_{2})_{1}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [274, 479, 291, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [291, 479, 326, 495], "score": 0.89, "content": "(2\\Lambda_{2})_{2}", "type": "inline_equation", "height": 16, "width": 35}], "index": 23}, {"bbox": [66, 497, 544, 516], "spans": [{"bbox": [66, 497, 82, 511], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [82, 497, 97, 516], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [98, 498, 114, 511], "score": 0.9, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [114, 497, 129, 516], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [130, 497, 180, 513], "score": 0.94, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 50}, {"bbox": [181, 497, 197, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [197, 497, 246, 513], "score": 0.91, "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "type": "inline_equation", "height": 16, "width": 49}, {"bbox": [246, 497, 262, 516], "score": 1.0, "content": " +", "type": "text"}, {"bbox": [263, 497, 321, 513], "score": 0.87, "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [322, 497, 338, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [338, 497, 398, 513], "score": 0.89, "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [398, 497, 414, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [414, 497, 491, 513], "score": 0.89, "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [492, 497, 509, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [509, 497, 544, 513], "score": 0.89, "content": "(3\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 35}], "index": 24}], "index": 23.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [66, 479, 544, 516]}, {"type": "text", "bbox": [70, 525, 542, 556], "lines": [{"bbox": [91, 522, 543, 548], "spans": [{"bbox": [91, 522, 408, 548], "score": 1.0, "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for", "type": "text"}, {"bbox": [409, 525, 432, 543], "score": 0.92, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [433, 522, 455, 548], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [455, 527, 490, 543], "score": 0.91, "content": "(\\pi_{3})^{\\pm1}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [491, 522, 506, 548], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [506, 528, 537, 541], "score": 0.87, "content": "k=3", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [537, 522, 543, 548], "score": 1.0, "content": ",", "type": "text"}], "index": 25}, {"bbox": [72, 543, 160, 557], "spans": [{"bbox": [72, 544, 93, 557], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [93, 544, 107, 556], "score": 0.84, "content": "\\pi_{4}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [108, 544, 124, 557], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [124, 543, 155, 555], "score": 0.89, "content": "k=4", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [155, 544, 160, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [72, 522, 543, 557]}, {"type": "title", "bbox": [249, 583, 362, 598], "lines": [{"bbox": [249, 585, 362, 599], "spans": [{"bbox": [249, 585, 362, 599], "score": 1.0, "content": "4. The Arguments", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 612, 541, 669], "lines": [{"bbox": [95, 614, 540, 628], "spans": [{"bbox": [95, 614, 540, 628], "score": 1.0, "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "type": "text"}], "index": 28}, {"bbox": [69, 628, 541, 642], "spans": [{"bbox": [69, 628, 541, 642], "score": 1.0, "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "type": "text"}], "index": 29}, {"bbox": [70, 641, 541, 659], "spans": [{"bbox": [70, 641, 541, 659], "score": 1.0, "content": "elements of finite order. These values have been studied by a number of people \u2014 see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] \u2014 and the resulting combinatorics is often quite pretty.", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [69, 614, 541, 673]}, {"type": "text", "bbox": [69, 670, 540, 713], "lines": [{"bbox": [93, 671, 540, 686], "spans": [{"bbox": [93, 671, 325, 686], "score": 1.0, "content": "Lemma 2.2 implies that a fusion-symmetry ", "type": "text"}, {"bbox": [325, 677, 332, 682], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [333, 671, 464, 686], "score": 1.0, "content": " preserves q-dimensions: ", "type": "text"}, {"bbox": [464, 673, 540, 685], "score": 0.94, "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "type": "inline_equation", "height": 12, "width": 76}], "index": 32}, {"bbox": [70, 686, 540, 701], "spans": [{"bbox": [70, 688, 116, 699], "score": 0.92, "content": "\\forall\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [116, 686, 373, 701], "score": 1.0, "content": ". In this subsection we use that to find a weight ", "type": "text"}, {"bbox": [374, 688, 387, 698], "score": 0.92, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [388, 686, 540, 701], "score": 1.0, "content": "for each algebra which must", "type": "text"}], "index": 33}, {"bbox": [70, 700, 203, 715], "spans": [{"bbox": [70, 700, 190, 715], "score": 1.0, "content": "be essentially fixed by ", "type": "text"}, {"bbox": [190, 705, 198, 711], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [198, 700, 203, 715], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 33, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [70, 671, 540, 715]}]}
0002044v1	15	4.1. $$q$$ -dimensions The most basic properties obeyed by the q-dimensions $$\begin{array}{r}{\mathcal{D}(\lambda)=\frac{S_{\lambda0}}{S_{00}}}\end{array}$$ are that $$\mathcal{D}(\lambda)\geq1$$ , and $$\mathcal{D}(s\lambda)=\mathcal{D}(\lambda)$$ for any $$s\in S$$ . Recall that $$\boldsymbol{S}$$ is the symmetry group of the extended Dynkin diagram of $$X_{r}^{(1)}$$ , and that $$s\in S$$ acts on $$P_{+}$$ by permuting the Dynkin labels. The argument yielding Proposition 4.1 below relies heavily on the following observa- tion. Use (2.1c) to extend the domain of $$\mathcal{D}$$ from $$P_{+}$$ to the fundamental chamber $$C_{+}$$ : Choose any $$a,b\in C_{+}$$ . Then a straightforward calculation from (2.1c) gives for $$0<t<1$$ . This means that for all $$0<t<1$$ , Proposition 4.1 [17,18]. For the following algebras $$X_{r}^{(1)}$$ and levels $$k$$ , and choices of weight $$\Lambda_{\star}$$ , $$\mathcal{D}(\lambda)=\mathcal{D}(\Lambda_{\star})$$ implies $$\lambda\in{\mathcal{S}}\Lambda_{\star}$$ : (a) For A(r1) any level $$k$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (b) For Br(1) any level $$k\neq2$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (c) For Cr(1) any level $$k$$ (except for $$(r,k)=(2,3)$$ or $$(3,2).$$ ), where $$\Lambda_{\star}=\Lambda_{1}$$ ; (d) For $$D_{r}^{(1)}$$ any level $$k\neq2$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (e6) For E6(1) any level $$k$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (e7) For $${E}_{7}^{(1)}$$ any level $$k\neq3$$ , where $$\Lambda_{\star}=\Lambda_{6}$$ ; (e8) For E8(1) any level $$k\neq1,4$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (f4) For $${F}_{4}^{(1)}$$ any level $$k\neq3,4$$ , where $$\Lambda_{\star}=\Lambda_{4}$$ ; (g2) For G(21) level any $$k\neq3,4$$ , where $$\Lambda_{\star}=\Lambda_{2}$$ . The missing cases are: $$B_{r,2}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})=\cdots={\mathcal{D}}(\Lambda_{r-1})={\mathcal{D}}(2\Lambda_{r});$$ $$D_{r,2}$$ where $${\mathcal{D}}(\Lambda_{1})=\cdots={\mathcal{D}}(\Lambda_{r-2})$$ ; $$C_{2,3}$$ where $${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(3\Lambda_{1})={\mathcal{D}}(\Lambda_{1})$$ , and its rank-level dual $$C_{3,2}$$ ; $$E_{7,3}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{6})$$ ; $$E_{8,1}$$ where $$\Lambda_{1}\notin P_{+}=\{0\}$$ , and $$E_{8,4}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{6})$$ ; $$F_{4,3}$$ where $${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{4})$$ , and $$F_{4,4}$$ where $$\mathcal{D}(\Lambda_{1})=\mathcal{D}(2\Lambda_{1})=\mathcal{D}(4\Lambda_{4})=\mathcal{D}(\Lambda_{4})$$ ; $$G_{2,3}$$ where $$\mathcal{D}(\Lambda_{1})=\mathcal{D}(\Lambda_{2})=\mathcal{D}(3\Lambda_{2})$$ , and $$G_{2,4}$$ where $$\mathcal{D}(\Lambda_{2})=\mathcal{D}(2\Lambda_{1})$$ . The weight $$\Lambda_{\star}$$ singled out by Proposition 4.1 (i.e. $$\Lambda_{\star}=\Lambda_{1}$$ for $$A_{r}^{(1)}$$ , ..., $$\Lambda_{\star}=\Lambda_{2}$$ for $$G_{2}^{(1)})$$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $$\Lambda_{\star}$$ will also have the smallest q-dimension after the simple-currents.	<p>4.1. $$q$$ -dimensions</p> <p>The most basic properties obeyed by the q-dimensions $$\begin{array}{r}{\mathcal{D}(\lambda)=\frac{S_{\lambda0}}{S_{00}}}\end{array}$$ are that $$\mathcal{D}(\lambda)\geq1$$ , and $$\mathcal{D}(s\lambda)=\mathcal{D}(\lambda)$$ for any $$s\in S$$ . Recall that $$\boldsymbol{S}$$ is the symmetry group of the extended Dynkin diagram of $$X_{r}^{(1)}$$ , and that $$s\in S$$ acts on $$P_{+}$$ by permuting the Dynkin labels.</p> <p>The argument yielding Proposition 4.1 below relies heavily on the following observa- tion. Use (2.1c) to extend the domain of $$\mathcal{D}$$ from $$P_{+}$$ to the fundamental chamber $$C_{+}$$ :</p> <p>Choose any $$a,b\in C_{+}$$ . Then a straightforward calculation from (2.1c) gives</p> <p>for $$0<t<1$$ . This means that for all $$0<t<1$$ ,</p> <p>Proposition 4.1 [17,18]. For the following algebras $$X_{r}^{(1)}$$ and levels $$k$$ , and choices of weight $$\Lambda_{\star}$$ , $$\mathcal{D}(\lambda)=\mathcal{D}(\Lambda_{\star})$$ implies $$\lambda\in{\mathcal{S}}\Lambda_{\star}$$ :</p> <p>(a) For A(r1) any level $$k$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (b) For Br(1) any level $$k\neq2$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (c) For Cr(1) any level $$k$$ (except for $$(r,k)=(2,3)$$ or $$(3,2).$$ ), where $$\Lambda_{\star}=\Lambda_{1}$$ ; (d) For $$D_{r}^{(1)}$$ any level $$k\neq2$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (e6) For E6(1) any level $$k$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (e7) For $${E}_{7}^{(1)}$$ any level $$k\neq3$$ , where $$\Lambda_{\star}=\Lambda_{6}$$ ; (e8) For E8(1) any level $$k\neq1,4$$ , where $$\Lambda_{\star}=\Lambda_{1}$$ ; (f4) For $${F}_{4}^{(1)}$$ any level $$k\neq3,4$$ , where $$\Lambda_{\star}=\Lambda_{4}$$ ; (g2) For G(21) level any $$k\neq3,4$$ , where $$\Lambda_{\star}=\Lambda_{2}$$ .</p> <p>The missing cases are: $$B_{r,2}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})=\cdots={\mathcal{D}}(\Lambda_{r-1})={\mathcal{D}}(2\Lambda_{r});$$ $$D_{r,2}$$ where $${\mathcal{D}}(\Lambda_{1})=\cdots={\mathcal{D}}(\Lambda_{r-2})$$ ; $$C_{2,3}$$ where $${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(3\Lambda_{1})={\mathcal{D}}(\Lambda_{1})$$ , and its rank-level dual $$C_{3,2}$$ ; $$E_{7,3}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{6})$$ ; $$E_{8,1}$$ where $$\Lambda_{1}\notin P_{+}=\{0\}$$ , and $$E_{8,4}$$ where $${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{6})$$ ; $$F_{4,3}$$ where $${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{4})$$ , and $$F_{4,4}$$ where $$\mathcal{D}(\Lambda_{1})=\mathcal{D}(2\Lambda_{1})=\mathcal{D}(4\Lambda_{4})=\mathcal{D}(\Lambda_{4})$$ ; $$G_{2,3}$$ where $$\mathcal{D}(\Lambda_{1})=\mathcal{D}(\Lambda_{2})=\mathcal{D}(3\Lambda_{2})$$ , and $$G_{2,4}$$ where $$\mathcal{D}(\Lambda_{2})=\mathcal{D}(2\Lambda_{1})$$ .</p> <p>The weight $$\Lambda_{\star}$$ singled out by Proposition 4.1 (i.e. $$\Lambda_{\star}=\Lambda_{1}$$ for $$A_{r}^{(1)}$$ , ..., $$\Lambda_{\star}=\Lambda_{2}$$ for $$G_{2}^{(1)})$$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $$\Lambda_{\star}$$ will also have the smallest q-dimension after the simple-currents.</p>	[{"type": "text", "coordinates": [71, 71, 167, 86], "content": "4.1. $$q$$ -dimensions", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 92, 541, 138], "content": "The most basic properties obeyed by the q-dimensions $$\\begin{array}{r}{\\mathcal{D}(\\lambda)=\\frac{S_{\\lambda0}}{S_{00}}}\\end{array}$$ are that $$\\mathcal{D}(\\lambda)\\geq1$$ ,\nand $$\\mathcal{D}(s\\lambda)=\\mathcal{D}(\\lambda)$$ for any $$s\\in S$$ . Recall that $$\\boldsymbol{S}$$ is the symmetry group of the extended\nDynkin diagram of $$X_{r}^{(1)}$$ , and that $$s\\in S$$ acts on $$P_{+}$$ by permuting the Dynkin labels.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 138, 541, 167], "content": "The argument yielding Proposition 4.1 below relies heavily on the following observa-\ntion. Use (2.1c) to extend the domain of $$\\mathcal{D}$$ from $$P_{+}$$ to the fundamental chamber $$C_{+}$$ :", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [173, 181, 438, 218], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [70, 228, 468, 244], "content": "Choose any $$a,b\\in C_{+}$$ . Then a straightforward calculation from (2.1c) gives", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [159, 257, 453, 286], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [70, 296, 324, 311], "content": "for $$0<t<1$$ . This means that for all $$0<t<1$$ ,", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [207, 325, 404, 340], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [71, 360, 542, 390], "content": "Proposition 4.1 [17,18]. For the following algebras $$X_{r}^{(1)}$$ and levels $$k$$ , and choices\nof weight $$\\Lambda_{\\star}$$ , $$\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})$$ implies $$\\lambda\\in{\\mathcal{S}}\\Lambda_{\\star}$$ :", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 392, 479, 542], "content": "(a) For A(r1) any level $$k$$ , where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(b) For Br(1) any level $$k\\neq2$$ , where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(c) For Cr(1) any level $$k$$ (except for $$(r,k)=(2,3)$$ or $$(3,2).$$ ), where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(d) For $$D_{r}^{(1)}$$ any level $$k\\neq2$$ , where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(e6) For E6(1) any level $$k$$ , where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(e7) For $${E}_{7}^{(1)}$$ any level $$k\\neq3$$ , where $$\\Lambda_{\\star}=\\Lambda_{6}$$ ;\n(e8) For E8(1) any level $$k\\neq1,4$$ , where $$\\Lambda_{\\star}=\\Lambda_{1}$$ ;\n(f4) For $${F}_{4}^{(1)}$$ any level $$k\\neq3,4$$ , where $$\\Lambda_{\\star}=\\Lambda_{4}$$ ;\n(g2) For G(21) level any $$k\\neq3,4$$ , where $$\\Lambda_{\\star}=\\Lambda_{2}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [93, 547, 521, 650], "content": "The missing cases are: $$B_{r,2}$$ where $${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});$$\n$$D_{r,2}$$ where $${\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})$$ ;\n$$C_{2,3}$$ where $${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})$$ , and its rank-level dual $$C_{3,2}$$ ;\n$$E_{7,3}$$ where $${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})$$ ;\n$$E_{8,1}$$ where $$\\Lambda_{1}\\notin P_{+}=\\{0\\}$$ , and $$E_{8,4}$$ where $${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})$$ ;\n$$F_{4,3}$$ where $${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})$$ , and $$F_{4,4}$$ where $$\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})$$ ;\n$$G_{2,3}$$ where $$\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})$$ , and $$G_{2,4}$$ where $$\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 654, 541, 715], "content": "The weight $$\\Lambda_{\\star}$$ singled out by Proposition 4.1 (i.e. $$\\Lambda_{\\star}=\\Lambda_{1}$$ for $$A_{r}^{(1)}$$ , ..., $$\\Lambda_{\\star}=\\Lambda_{2}$$ for\n$$G_{2}^{(1)})$$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all\nbut the smallest levels (see [18, Table 3]), $$\\Lambda_{\\star}$$ will also have the smallest q-dimension after\nthe simple-currents.", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [70, 74, 97, 87], "content": "4.1. ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [97, 79, 102, 87], "content": "q", "score": 0.35, "index": 2}, {"type": "text", "coordinates": [103, 74, 167, 87], "content": "-dimensions", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [92, 91, 379, 112], "content": "The most basic properties obeyed by the q-dimensions ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [380, 95, 440, 111], "content": "\\begin{array}{r}{\\mathcal{D}(\\lambda)=\\frac{S_{\\lambda0}}{S_{00}}}\\end{array}", "score": 0.96, "index": 5}, {"type": "text", "coordinates": [440, 91, 488, 112], "content": " are that ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [488, 96, 536, 109], "content": "\\mathcal{D}(\\lambda)\\geq1", "score": 0.94, "index": 7}, {"type": "text", "coordinates": [537, 91, 542, 112], "content": ",", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [70, 108, 95, 125], "content": "and ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [95, 110, 170, 123], "content": "\\mathcal{D}(s\\lambda)=\\mathcal{D}(\\lambda)", "score": 0.94, "index": 10}, {"type": "text", "coordinates": [170, 108, 216, 125], "content": " for any ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [217, 111, 246, 120], "content": "s\\in S", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [247, 108, 319, 125], "content": ". Recall that ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [320, 111, 328, 120], "content": "\\boldsymbol{S}", "score": 0.89, "index": 14}, {"type": "text", "coordinates": [329, 108, 541, 125], "content": " is the symmetry group of the extended", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [70, 124, 173, 142], "content": "Dynkin diagram of ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [174, 123, 198, 138], "content": "X_{r}^{(1)}", "score": 0.93, "index": 17}, {"type": "text", "coordinates": [198, 124, 254, 142], "content": ", and that ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [254, 128, 282, 137], "content": "s\\in S", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [283, 124, 327, 142], "content": " acts on ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [327, 128, 343, 140], "content": "P_{+}", "score": 0.93, "index": 21}, {"type": "text", "coordinates": [343, 124, 519, 142], "content": " by permuting the Dynkin labels.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [93, 140, 540, 156], "content": "The argument yielding Proposition 4.1 below relies heavily on the following observa-", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [71, 155, 287, 169], "content": "tion. Use (2.1c) to extend the domain of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [288, 156, 298, 165], "content": "\\mathcal{D}", "score": 0.91, "index": 25}, {"type": "text", "coordinates": [298, 155, 329, 169], "content": " from ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [329, 156, 345, 168], "content": "P_{+}", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [345, 155, 501, 169], "content": " to the fundamental chamber ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [501, 156, 518, 168], "content": "C_{+}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [518, 155, 522, 169], "content": ":", "score": 1.0, "index": 30}, {"type": "interline_equation", "coordinates": [173, 181, 438, 218], "content": "C_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left\|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .", "score": 0.94, "index": 31}, {"type": "text", "coordinates": [71, 231, 135, 246], "content": "Choose any ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [136, 234, 183, 245], "content": "a,b\\in C_{+}", "score": 0.94, "index": 33}, {"type": "text", "coordinates": [184, 231, 467, 246], "content": ". Then a straightforward calculation from (2.1c) gives", "score": 1.0, "index": 34}, {"type": "interline_equation", "coordinates": [159, 257, 453, 286], "content": "\\frac{d}{d t}\\mathcal{D}(t a+(1-t)b)=0\\quad\\Longrightarrow\\quad\\frac{d^{2}}{d t^{2}}\\mathcal{D}(t a+(1-t)b)<0", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [70, 298, 89, 313], "content": "for ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [90, 301, 138, 309], "content": "0<t<1", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [138, 298, 271, 313], "content": ". This means that for all ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [271, 301, 320, 310], "content": "0<t<1", "score": 0.88, "index": 39}, {"type": "text", "coordinates": [320, 298, 324, 313], "content": ",", "score": 1.0, "index": 40}, {"type": "interline_equation", "coordinates": [207, 325, 404, 340], "content": "{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.", "score": 0.88, "index": 41}, {"type": "text", "coordinates": [91, 360, 380, 380], "content": "Proposition 4.1 [17,18]. For the following algebras ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [380, 360, 406, 376], "content": "X_{r}^{(1)}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [406, 360, 464, 380], "content": "and levels ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [464, 364, 472, 375], "content": "k", "score": 0.79, "index": 45}, {"type": "text", "coordinates": [472, 360, 543, 380], "content": ", and choices", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [71, 378, 122, 394], "content": "of weight ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [122, 380, 136, 391], "content": "\\Lambda_{\\star}", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [137, 378, 144, 394], "content": ", ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [144, 379, 218, 392], "content": "\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})", "score": 0.92, "index": 50}, {"type": "text", "coordinates": [219, 378, 263, 394], "content": " implies ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [263, 378, 308, 390], "content": "\\lambda\\in{\\mathcal{S}}\\Lambda_{\\star}", "score": 0.9, "index": 52}, {"type": "text", "coordinates": [308, 378, 314, 394], "content": ":", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [72, 392, 142, 409], "content": "(a) For A(r1)", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [142, 394, 192, 409], "content": "any level ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [193, 396, 200, 406], "content": "k", "score": 0.77, "index": 56}, {"type": "text", "coordinates": [201, 394, 241, 409], "content": ", where ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [241, 394, 285, 408], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [286, 394, 290, 409], "content": ";", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [72, 409, 142, 425], "content": "(b) For Br(1)", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [142, 412, 193, 425], "content": "any level ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [194, 411, 223, 425], "content": "k\\neq2", "score": 0.91, "index": 62}, {"type": "text", "coordinates": [223, 412, 263, 425], "content": ", where ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [264, 411, 308, 424], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.91, "index": 64}, {"type": "text", "coordinates": [308, 412, 312, 425], "content": ";", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [72, 424, 144, 443], "content": "(c) For Cr(1)", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [143, 427, 193, 442], "content": "any level ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [193, 428, 201, 439], "content": "k", "score": 0.78, "index": 68}, {"type": "text", "coordinates": [202, 427, 264, 442], "content": " (except for ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [264, 427, 334, 442], "content": "(r,k)=(2,3)", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [334, 427, 353, 442], "content": " or ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [353, 428, 380, 442], "content": "(3,2).", "score": 0.52, "index": 72}, {"type": "text", "coordinates": [381, 427, 425, 442], "content": "), where ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [426, 428, 470, 441], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [470, 427, 474, 442], "content": ";", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [71, 444, 117, 458], "content": "(d) For ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [117, 442, 141, 457], "content": "D_{r}^{(1)}", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [143, 445, 194, 459], "content": "any level ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [194, 444, 224, 458], "content": "k\\neq2", "score": 0.91, "index": 79}, {"type": "text", "coordinates": [224, 445, 264, 459], "content": ", where ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [264, 444, 308, 458], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [309, 445, 314, 459], "content": ";", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [65, 457, 144, 477], "content": "(e6) For E6(1)", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [143, 462, 193, 476], "content": "any level ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [193, 462, 201, 473], "content": "k", "score": 0.78, "index": 85}, {"type": "text", "coordinates": [201, 462, 241, 476], "content": ", where ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [242, 461, 286, 474], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [286, 462, 291, 476], "content": ";", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [67, 475, 117, 493], "content": "(e7) For ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [117, 475, 140, 492], "content": "{E}_{7}^{(1)}", "score": 0.89, "index": 90}, {"type": "text", "coordinates": [144, 478, 193, 493], "content": "any level ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [193, 478, 223, 492], "content": "k\\neq3", "score": 0.91, "index": 92}, {"type": "text", "coordinates": [223, 478, 263, 493], "content": ", where ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [264, 478, 308, 491], "content": "\\Lambda_{\\star}=\\Lambda_{6}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [308, 478, 312, 493], "content": ";", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [63, 489, 146, 516], "content": "(e8) For E8(1)", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [143, 495, 193, 509], "content": "any level ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [193, 495, 235, 509], "content": "k\\neq1,4", "score": 0.9, "index": 98}, {"type": "text", "coordinates": [235, 495, 275, 509], "content": ", where ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [275, 495, 319, 508], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "score": 0.92, "index": 100}, {"type": "text", "coordinates": [320, 495, 324, 509], "content": ";", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [70, 511, 117, 526], "content": "(f4) For ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [117, 510, 140, 526], "content": "{F}_{4}^{(1)}", "score": 0.91, "index": 103}, {"type": "text", "coordinates": [143, 512, 193, 526], "content": "any level ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [194, 513, 235, 525], "content": "k\\neq3,4", "score": 0.89, "index": 105}, {"type": "text", "coordinates": [235, 512, 274, 526], "content": ", where ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [275, 511, 319, 525], "content": "\\Lambda_{\\star}=\\Lambda_{4}", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [320, 512, 324, 526], "content": ";", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [64, 523, 146, 549], "content": "(g2) For G(21)", "score": 1.0, "index": 109}, {"type": "text", "coordinates": [142, 527, 193, 544], "content": "level any ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [194, 529, 235, 542], "content": "k\\neq3,4", "score": 0.89, "index": 111}, {"type": "text", "coordinates": [235, 527, 274, 544], "content": ", where ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [275, 528, 319, 542], "content": "\\Lambda_{\\star}=\\Lambda_{2}", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [320, 527, 324, 544], "content": ".", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [94, 550, 216, 566], "content": "The missing cases are: ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [217, 551, 239, 564], "content": "B_{r,2}", "score": 0.89, "index": 116}, {"type": "text", "coordinates": [239, 550, 276, 566], "content": " where ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [277, 551, 505, 564], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});", "score": 0.87, "index": 118}, {"type": "inline_equation", "coordinates": [95, 567, 117, 579], "content": "D_{r,2}", "score": 0.92, "index": 119}, {"type": "text", "coordinates": [118, 564, 155, 580], "content": " where ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [156, 565, 279, 578], "content": "{\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})", "score": 0.91, "index": 121}, {"type": "text", "coordinates": [280, 564, 285, 580], "content": ";", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [95, 581, 117, 593], "content": "C_{2,3}", "score": 0.92, "index": 123}, {"type": "text", "coordinates": [117, 578, 155, 596], "content": " where ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [155, 580, 290, 593], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})", "score": 0.92, "index": 125}, {"type": "text", "coordinates": [291, 578, 419, 596], "content": ", and its rank-level dual ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [419, 579, 441, 593], "content": "C_{3,2}", "score": 0.9, "index": 127}, {"type": "text", "coordinates": [441, 578, 446, 596], "content": ";", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [95, 595, 117, 608], "content": "E_{7,3}", "score": 0.92, "index": 129}, {"type": "text", "coordinates": [117, 593, 155, 608], "content": " where ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [155, 594, 285, 607], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})", "score": 0.92, "index": 131}, {"type": "text", "coordinates": [286, 593, 290, 608], "content": ";", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [95, 610, 117, 622], "content": "E_{8,1}", "score": 0.92, "index": 133}, {"type": "text", "coordinates": [117, 607, 155, 622], "content": " where ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [155, 609, 234, 622], "content": "\\Lambda_{1}\\notin P_{+}=\\{0\\}", "score": 0.92, "index": 135}, {"type": "text", "coordinates": [234, 607, 263, 622], "content": ", and ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [264, 610, 286, 622], "content": "E_{8,4}", "score": 0.93, "index": 137}, {"type": "text", "coordinates": [286, 607, 324, 622], "content": " where ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [324, 609, 405, 621], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})", "score": 0.93, "index": 139}, {"type": "text", "coordinates": [406, 607, 410, 622], "content": ";", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [95, 624, 115, 636], "content": "F_{4,3}", "score": 0.92, "index": 141}, {"type": "text", "coordinates": [116, 622, 154, 637], "content": " where ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [155, 623, 236, 636], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})", "score": 0.92, "index": 143}, {"type": "text", "coordinates": [236, 622, 265, 637], "content": ", and ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [266, 624, 286, 636], "content": "F_{4,4}", "score": 0.92, "index": 145}, {"type": "text", "coordinates": [287, 622, 324, 637], "content": " where ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [325, 623, 516, 636], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [516, 622, 519, 637], "content": ";", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [95, 639, 118, 651], "content": "G_{2,3}", "score": 0.92, "index": 149}, {"type": "text", "coordinates": [118, 636, 155, 653], "content": " where ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [156, 638, 292, 650], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})", "score": 0.91, "index": 151}, {"type": "text", "coordinates": [293, 636, 321, 653], "content": ", and ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [322, 638, 344, 651], "content": "G_{2,4}", "score": 0.91, "index": 153}, {"type": "text", "coordinates": [345, 636, 383, 653], "content": " where ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [383, 637, 470, 650], "content": "\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})", "score": 0.92, "index": 155}, {"type": "text", "coordinates": [470, 636, 475, 653], "content": ".", "score": 1.0, "index": 156}, {"type": "text", "coordinates": [92, 654, 158, 673], "content": "The weight ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [158, 659, 172, 669], "content": "\\Lambda_{\\star}", "score": 0.91, "index": 158}, {"type": "text", "coordinates": [172, 654, 362, 673], "content": "singled out by Proposition 4.1 (i.e. ", "score": 1.0, "index": 159}, {"type": 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"content": "k\\neq3,4", "caption": ""}, {"type": "inline", "coordinates": [275, 511, 319, 525], "content": "\\Lambda_{\\star}=\\Lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [194, 529, 235, 542], "content": "k\\neq3,4", "caption": ""}, {"type": "inline", "coordinates": [275, 528, 319, 542], "content": "\\Lambda_{\\star}=\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [217, 551, 239, 564], "content": "B_{r,2}", "caption": ""}, {"type": "inline", "coordinates": [277, 551, 505, 564], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});", "caption": ""}, {"type": "inline", "coordinates": [95, 567, 117, 579], "content": "D_{r,2}", "caption": ""}, {"type": "inline", "coordinates": [156, 565, 279, 578], "content": "{\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})", "caption": ""}, {"type": "inline", "coordinates": [95, 581, 117, 593], "content": "C_{2,3}", "caption": ""}, {"type": "inline", "coordinates": [155, 580, 290, 593], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [419, 579, 441, 593], "content": "C_{3,2}", "caption": ""}, {"type": "inline", "coordinates": [95, 595, 117, 608], "content": "E_{7,3}", "caption": ""}, {"type": "inline", "coordinates": [155, 594, 285, 607], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})", "caption": ""}, {"type": "inline", "coordinates": [95, 610, 117, 622], "content": "E_{8,1}", "caption": ""}, {"type": "inline", "coordinates": [155, 609, 234, 622], "content": "\\Lambda_{1}\\notin P_{+}=\\{0\\}", "caption": ""}, {"type": "inline", "coordinates": [264, 610, 286, 622], "content": "E_{8,4}", "caption": ""}, {"type": "inline", "coordinates": [324, 609, 405, 621], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})", "caption": ""}, {"type": "inline", "coordinates": [95, 624, 115, 636], "content": "F_{4,3}", "caption": ""}, {"type": "inline", "coordinates": [155, 623, 236, 636], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})", "caption": ""}, {"type": "inline", "coordinates": [266, 624, 286, 636], "content": "F_{4,4}", "caption": ""}, {"type": "inline", "coordinates": [325, 623, 516, 636], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})", "caption": ""}, {"type": "inline", "coordinates": [95, 639, 118, 651], "content": "G_{2,3}", "caption": ""}, {"type": "inline", "coordinates": [156, 638, 292, 650], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})", "caption": ""}, {"type": "inline", "coordinates": [322, 638, 344, 651], "content": "G_{2,4}", "caption": ""}, {"type": "inline", "coordinates": [383, 637, 470, 650], "content": "\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [158, 659, 172, 669], "content": "\\Lambda_{\\star}", "caption": ""}, {"type": "inline", "coordinates": [362, 659, 407, 669], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [429, 655, 451, 669], "content": "A_{r}^{(1)}", "caption": ""}, {"type": "inline", "coordinates": [477, 659, 521, 669], "content": "\\Lambda_{\\star}=\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [71, 671, 97, 687], "content": "G_{2}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [291, 690, 305, 700], "content": "\\Lambda_{\\star}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "4.1. $q$ -dimensions ", "page_idx": 15}, {"type": "text", "text": "The most basic properties obeyed by the q-dimensions $\\begin{array}{r}{\\mathcal{D}(\\lambda)=\\frac{S_{\\lambda0}}{S_{00}}}\\end{array}$ are that $\\mathcal{D}(\\lambda)\\geq1$ , and $\\mathcal{D}(s\\lambda)=\\mathcal{D}(\\lambda)$ for any $s\\in S$ . Recall that $\\boldsymbol{S}$ is the symmetry group of the extended Dynkin diagram of $X_{r}^{(1)}$ , and that $s\\in S$ acts on $P_{+}$ by permuting the Dynkin labels. ", "page_idx": 15}, {"type": "text", "text": "The argument yielding Proposition 4.1 below relies heavily on the following observation. Use (2.1c) to extend the domain of $\\mathcal{D}$ from $P_{+}$ to the fundamental chamber $C_{+}$ : ", "page_idx": 15}, {"type": "equation", "text": "$$\nC_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left\|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "Choose any $a,b\\in C_{+}$ . Then a straightforward calculation from (2.1c) gives ", "page_idx": 15}, {"type": "equation", "text": "$$\n\\frac{d}{d t}\\mathcal{D}(t a+(1-t)b)=0\\quad\\Longrightarrow\\quad\\frac{d^{2}}{d t^{2}}\\mathcal{D}(t a+(1-t)b)<0\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "for $0<t<1$ . This means that for all $0<t<1$ , ", "page_idx": 15}, {"type": "equation", "text": "$$\n{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "Proposition 4.1 [17,18]. For the following algebras $X_{r}^{(1)}$ and levels $k$ , and choices of weight $\\Lambda_{\\star}$ , $\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})$ implies $\\lambda\\in{\\mathcal{S}}\\Lambda_{\\star}$ : ", "page_idx": 15}, {"type": "text", "text": "(a) For A(r1) any level $k$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (b) For Br(1) any level $k\\neq2$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (c) For Cr(1) any level $k$ (except for $(r,k)=(2,3)$ or $(3,2).$ ), where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (d) For $D_{r}^{(1)}$ any level $k\\neq2$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (e6) For E6(1) any level $k$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (e7) For ${E}_{7}^{(1)}$ any level $k\\neq3$ , where $\\Lambda_{\\star}=\\Lambda_{6}$ ; (e8) For E8(1) any level $k\\neq1,4$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (f4) For ${F}_{4}^{(1)}$ any level $k\\neq3,4$ , where $\\Lambda_{\\star}=\\Lambda_{4}$ ; (g2) For G(21) level any $k\\neq3,4$ , where $\\Lambda_{\\star}=\\Lambda_{2}$ . ", "page_idx": 15}, {"type": "text", "text": "The missing cases are: $B_{r,2}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});$ $D_{r,2}$ where ${\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})$ ; \n$C_{2,3}$ where ${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})$ , and its rank-level dual $C_{3,2}$ ; \n$E_{7,3}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})$ ; \n$E_{8,1}$ where $\\Lambda_{1}\\notin P_{+}=\\{0\\}$ , and $E_{8,4}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})$ ; \n$F_{4,3}$ where ${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})$ , and $F_{4,4}$ where $\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})$ ; \n$G_{2,3}$ where $\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})$ , and $G_{2,4}$ where $\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})$ . ", "page_idx": 15}, {"type": "text", "text": "The weight $\\Lambda_{\\star}$ singled out by Proposition 4.1 (i.e. $\\Lambda_{\\star}=\\Lambda_{1}$ for $A_{r}^{(1)}$ , ..., $\\Lambda_{\\star}=\\Lambda_{2}$ for $G_{2}^{(1)})$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $\\Lambda_{\\star}$ will also have the smallest q-dimension after the simple-currents. 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Recall that ", "type": "text"}, {"bbox": [320, 111, 328, 120], "score": 0.89, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [329, 108, 541, 125], "score": 1.0, "content": " is the symmetry group of the extended", "type": "text"}], "index": 2}, {"bbox": [70, 123, 519, 142], "spans": [{"bbox": [70, 124, 173, 142], "score": 1.0, "content": "Dynkin diagram of ", "type": "text"}, {"bbox": [174, 123, 198, 138], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [198, 124, 254, 142], "score": 1.0, "content": ", and that ", "type": "text"}, {"bbox": [254, 128, 282, 137], "score": 0.92, "content": "s\\in S", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [283, 124, 327, 142], "score": 1.0, "content": " acts on ", "type": "text"}, {"bbox": [327, 128, 343, 140], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [343, 124, 519, 142], "score": 1.0, "content": " by permuting the Dynkin labels.", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [70, 91, 542, 142]}, {"type": "text", "bbox": [70, 138, 541, 167], "lines": [{"bbox": [93, 140, 540, 156], "spans": [{"bbox": [93, 140, 540, 156], "score": 1.0, "content": "The argument yielding Proposition 4.1 below relies heavily on the following observa-", "type": "text"}], "index": 4}, {"bbox": [71, 155, 522, 169], "spans": [{"bbox": [71, 155, 287, 169], "score": 1.0, "content": "tion. Use (2.1c) to extend the domain of ", "type": "text"}, {"bbox": [288, 156, 298, 165], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [298, 155, 329, 169], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [329, 156, 345, 168], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [345, 155, 501, 169], "score": 1.0, "content": " to the fundamental chamber ", "type": "text"}, {"bbox": [501, 156, 518, 168], "score": 0.92, "content": "C_{+}", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [518, 155, 522, 169], "score": 1.0, "content": ":", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [71, 140, 540, 169]}, {"type": "interline_equation", "bbox": [173, 181, 438, 218], "lines": [{"bbox": [173, 181, 438, 218], "spans": [{"bbox": [173, 181, 438, 218], "score": 0.94, "content": "C_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left\|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .", "type": "interline_equation"}], "index": 6}], "index": 6, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 228, 468, 244], "lines": [{"bbox": [71, 231, 467, 246], "spans": [{"bbox": [71, 231, 135, 246], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [136, 234, 183, 245], "score": 0.94, "content": "a,b\\in C_{+}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [184, 231, 467, 246], "score": 1.0, "content": ". Then a straightforward calculation from (2.1c) gives", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [71, 231, 467, 246]}, {"type": "interline_equation", "bbox": [159, 257, 453, 286], "lines": [{"bbox": [159, 257, 453, 286], "spans": [{"bbox": [159, 257, 453, 286], "score": 0.92, "content": "\\frac{d}{d t}\\mathcal{D}(t a+(1-t)b)=0\\quad\\Longrightarrow\\quad\\frac{d^{2}}{d t^{2}}\\mathcal{D}(t a+(1-t)b)<0", "type": "interline_equation"}], "index": 8}], "index": 8, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 296, 324, 311], "lines": [{"bbox": [70, 298, 324, 313], "spans": [{"bbox": [70, 298, 89, 313], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [90, 301, 138, 309], "score": 0.91, "content": "0<t<1", "type": "inline_equation", "height": 8, "width": 48}, {"bbox": [138, 298, 271, 313], "score": 1.0, "content": ". This means that for all ", "type": "text"}, {"bbox": [271, 301, 320, 310], "score": 0.88, "content": "0<t<1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [320, 298, 324, 313], "score": 1.0, "content": ",", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [70, 298, 324, 313]}, {"type": "interline_equation", "bbox": [207, 325, 404, 340], "lines": [{"bbox": [207, 325, 404, 340], "spans": [{"bbox": [207, 325, 404, 340], "score": 0.88, "content": "{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.", "type": "interline_equation"}], "index": 10}], "index": 10, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 360, 542, 390], "lines": [{"bbox": [91, 360, 543, 380], "spans": [{"bbox": [91, 360, 380, 380], "score": 1.0, "content": "Proposition 4.1 [17,18]. For the following algebras ", "type": "text"}, {"bbox": [380, 360, 406, 376], "score": 0.91, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 26}, {"bbox": [406, 360, 464, 380], "score": 1.0, "content": "and levels ", "type": "text"}, {"bbox": [464, 364, 472, 375], "score": 0.79, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [472, 360, 543, 380], "score": 1.0, "content": ", and choices", "type": "text"}], "index": 11}, {"bbox": [71, 378, 314, 394], "spans": [{"bbox": [71, 378, 122, 394], "score": 1.0, "content": "of weight ", "type": "text"}, {"bbox": [122, 380, 136, 391], "score": 0.89, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [137, 378, 144, 394], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [144, 379, 218, 392], "score": 0.92, "content": "\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": 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"text"}, {"bbox": [275, 528, 319, 542], "score": 0.92, "content": "\\Lambda_{\\star}=\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [320, 527, 324, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 17, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [63, 392, 474, 549]}, {"type": "list", "bbox": [93, 547, 521, 650], "lines": [{"bbox": [94, 550, 505, 566], "spans": [{"bbox": [94, 550, 216, 566], "score": 1.0, "content": "The missing cases are: ", "type": "text"}, {"bbox": [217, 551, 239, 564], "score": 0.89, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [239, 550, 276, 566], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [277, 551, 505, 564], "score": 0.87, "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});", "type": "inline_equation", "height": 13, "width": 228}], 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"width": 22}, {"bbox": [452, 654, 476, 673], "score": 1.0, "content": ", ..., ", "type": "text"}, {"bbox": [477, 659, 521, 669], "score": 0.86, "content": "\\Lambda_{\\star}=\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [521, 654, 542, 673], "score": 1.0, "content": " for", "type": "text"}], "index": 29}, {"bbox": [71, 669, 543, 691], "spans": [{"bbox": [71, 671, 97, 687], "score": 0.89, "content": "G_{2}^{(1)})", "type": "inline_equation", "height": 16, "width": 26}, {"bbox": [97, 669, 543, 691], "score": 1.0, "content": " is the nonzero weight with smallest Weyl dimension. What we find is that, for all", "type": "text"}], "index": 30}, {"bbox": [69, 687, 541, 703], "spans": [{"bbox": [69, 687, 290, 703], "score": 1.0, "content": "but the smallest levels (see [18, Table 3]), ", "type": "text"}, {"bbox": [291, 690, 305, 700], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 687, 541, 703], "score": 1.0, "content": "will also have the smallest q-dimension after", "type": "text"}], "index": 31}, {"bbox": [70, 702, 176, 718], "spans": [{"bbox": [70, 702, 176, 718], "score": 1.0, "content": "the simple-currents.", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [69, 654, 543, 718]}]}
0002044v1	16	The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $$(A_{r}^{(1)})$$ and the most difficult $$(E_{8}^{(1)})$$ cases. Consider first $$A_{r,k}$$ . By choosing $$a\!-\!b=\Lambda_{i}-\Lambda_{j}$$ in (4.1), we get that either $$\lambda=k\Lambda_{\ell}$$ for some $$\ell$$ , in which case $$\lambda$$ is a simple-current and (for $$k\neq1$$ ) $$D(\lambda)<\mathcal{D}(\Lambda_{1})$$ , or $$\mathcal{D}(\lambda)\ge\mathcal{D}(\Lambda_{\ell})$$ for some $$\ell$$ , with equality iff $$\lambda\in S\Lambda_{\ell}$$ . But then rank-level duality $$A_{r,k}\leftrightarrow A_{k-1,r+1}$$ (defined as for $$C_{r,k}$$ , and which is exact for $$A_{r,k}$$ q-dimensions) and (4.1) with $$a-b=\widetilde{\Lambda_{0}}-\widetilde{\Lambda_{1}}$$ give us $$\mathcal{D}(\Lambda_{\ell})\,=\,\widetilde{\mathcal{D}}(\ell\widetilde{\Lambda_{1}})\,\geq\,\widetilde{\mathcal{D}}(\widetilde{\Lambda_{1}})\,=\,\mathcal{D}(\Lambda_{1})$$ , with equality iff $$\ell\,=\,1$$ or $$r$$ . Com bining these results yields Proposition 4.1(a). For $$E_{8,k}$$ , run through each $$a\mathrm{~-~}b\mathrm{~=~}a_{j}^{\vee}\Lambda_{i}\mathrm{~-~}a_{i}^{\vee}\Lambda_{j}$$ to reduce the proof to comparing $$\mathcal{D}(\Lambda_{1})$$ with $$\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$$ for $$i\neq0$$ , or $$\mathcal{D}(\Lambda_{i})$$ for $$i\neq0,1$$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $$\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$$ monotonically increases with $$k$$ to $$\infty$$ , while $$\mathcal{D}(\Lambda_{i})$$ monotonically increases with $$k$$ to the Weyl dimension of $$\Lambda_{i}$$ . The proof of Proposition 4.1(e8) then reduces to a short computation. # 4.2. The $$A$$ -series argument Recall that $$\overline{r}\,=\,r\,+\,1$$ . Proposition 4.1(a) tells us that $$\pi\Lambda_{1}\,=\,C^{a}J^{b}\Lambda_{1}$$ , for some $$a,b$$ . Hitting $$\pi$$ with $$C^{a}$$ , we can assume without loss of generality that $$a\;=\;0$$ . Write $$\pi(J0)=J^{c}0$$ ; then $$\pi$$ can be a permutation of $$P_{+}$$ only if $$c$$ is coprime to $$\overline{r}$$ . If $$k=1$$ then $$P_{+}=\{0,J0,\dots,J^{r}0\}$$ so $$\pi=\pi[c-1]$$ . Thus we can assume $$k\geq2$$ . Useful is the coefficient of $$\lambda$$ in the tensor product $$\Lambda_{1}\otimes\cdot\cdot\otimes\Lambda_{1}$$ ( $$\ell\,\mathrm{times})$$ : it is 0 unless $$t(\lambda)=\ell$$ , in which case the coefficient is $$\frac{\ell!}{h(\lambda)}$$ (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights $$\Lambda_{\overline{{r}}}$$ and $$\Lambda_{0}$$ , so e.g. $$^{\star}{\frac{k}{r}}\Lambda_{\overline{{r}}}^{\ \ \,}{}^{\ ,}$$ equals $$\mathrm{\Delta}^{\prime}0^{\circ}$$ when $$\overline{r}$$ divides $$k$$ . Here, $$h(\lambda)=\prod h(x)$$ is the product of the hook-lengths of the Young diagram corresponding to $$\lambda$$ . Equation (2.4) tells us that as long as $$t(\lambda)=\ell\leq k$$ , the number $$\frac{\ell!}{h(\lambda)}$$ will also be the coefficient of $$N_{\lambda}$$ in the fusion power $$(N_{\Lambda_{1}})^{\ell}$$ . Note that $$J0=k\Lambda_{1}$$ is the only simple-current appearing in the fusion product $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k$$ times). Thus the only nontrivial simple-current appearing in the fusion $$\pi\Lambda_{1}$$ × · · · × $$\pi\Lambda_{1}$$ will be $$J^{b k}J0$$ (0 will appear iff $$\overline{r}$$ divides $$k$$ ). Hence $$b k+1\equiv c$$ (mod $$\overline{r}$$ ) must be coprime to $$\overline{r}$$ . This is precisely the condition needed for $$\pi[b]$$ to be a simple-current automorphism. In other words, it suffices to consider $$\pi\Lambda_{1}=\Lambda_{1}$$ and hence $$\pi[J0]=J0$$ . We are done if $$r=1$$ , so assume $$r\geq2$$ . From the $$\Lambda_{1}$$ × $$\Lambda_{1}$$ fusion, we get that $$\pi\Lambda_{2}\in\{\Lambda_{2},2\Lambda_{1}\}$$ . Note that $$k\Lambda_{1}$$ occurs (with multiplicity 1) in the tensor and fusion product of $$2\Lambda_{1}$$ with $$k-2\;\Lambda_{1}\,{}^{\prime}\mathrm{s}$$ s, but that it doesn’t in the tensor (hence fusion) product of $$\Lambda_{2}$$ with $$k-2~\Lambda_{1}$$ ’s (recall that $$k\Lambda_{1}\succ(k-2)\Lambda_{1}+\Lambda_{2}$$ in the usual partial order on weights). Since $$\Lambda_{2}$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ does not contain $$J0$$ , $$(\pi\Lambda_{2})\boxtimes(\pi\Lambda_{1})\boxtimes\dots\boxtimes(\pi\Lambda_{1})$$ should also avoid $$\pi(J0)\,=\,J0$$ , and thus $$\pi\Lambda_{2}$$ cannot equal $$2\Lambda_{1}$$ . Thus we know $$\pi\Lambda_{2}=\Lambda_{2}$$ . The remaining $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ follow quickly from induction: if $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ for some $$2\leq\ell<r$$ , then the fusion $$\Lambda_{1}\boxtimes\Lambda_{\ell}$$ tells us $$\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$$ . But $$h(\Lambda_{1}+\Lambda_{\ell})=(\ell+1)!/\ell$$ and $$h(\Lambda_{\ell+1})=(\ell+1)!$$ , so $$\pi\Lambda_{\ell+1}=\Lambda_{\ell+1}$$ . Thus $$\pi$$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that $$\pi$$ must fix everything in $$P_{+}$$ .	<p>The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $$(A_{r}^{(1)})$$ and the most difficult $$(E_{8}^{(1)})$$ cases.</p> <p>Consider first $$A_{r,k}$$ . By choosing $$a\!-\!b=\Lambda_{i}-\Lambda_{j}$$ in (4.1), we get that either $$\lambda=k\Lambda_{\ell}$$ for some $$\ell$$ , in which case $$\lambda$$ is a simple-current and (for $$k\neq1$$ ) $$D(\lambda)<\mathcal{D}(\Lambda_{1})$$ , or $$\mathcal{D}(\lambda)\ge\mathcal{D}(\Lambda_{\ell})$$ for some $$\ell$$ , with equality iff $$\lambda\in S\Lambda_{\ell}$$ . But then rank-level duality $$A_{r,k}\leftrightarrow A_{k-1,r+1}$$ (defined as for $$C_{r,k}$$ , and which is exact for $$A_{r,k}$$ q-dimensions) and (4.1) with $$a-b=\widetilde{\Lambda_{0}}-\widetilde{\Lambda_{1}}$$ give us $$\mathcal{D}(\Lambda_{\ell})\,=\,\widetilde{\mathcal{D}}(\ell\widetilde{\Lambda_{1}})\,\geq\,\widetilde{\mathcal{D}}(\widetilde{\Lambda_{1}})\,=\,\mathcal{D}(\Lambda_{1})$$ , with equality iff $$\ell\,=\,1$$ or $$r$$ . Com bining these results yields Proposition 4.1(a).</p> <p>For $$E_{8,k}$$ , run through each $$a\mathrm{~-~}b\mathrm{~=~}a_{j}^{\vee}\Lambda_{i}\mathrm{~-~}a_{i}^{\vee}\Lambda_{j}$$ to reduce the proof to comparing $$\mathcal{D}(\Lambda_{1})$$ with $$\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$$ for $$i\neq0$$ , or $$\mathcal{D}(\Lambda_{i})$$ for $$i\neq0,1$$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $$\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$$ monotonically increases with $$k$$ to $$\infty$$ , while $$\mathcal{D}(\Lambda_{i})$$ monotonically increases with $$k$$ to the Weyl dimension of $$\Lambda_{i}$$ . The proof of Proposition 4.1(e8) then reduces to a short computation.</p> <h1>4.2. The $$A$$ -series argument</h1> <p>Recall that $$\overline{r}\,=\,r\,+\,1$$ . Proposition 4.1(a) tells us that $$\pi\Lambda_{1}\,=\,C^{a}J^{b}\Lambda_{1}$$ , for some $$a,b$$ . Hitting $$\pi$$ with $$C^{a}$$ , we can assume without loss of generality that $$a\;=\;0$$ . Write $$\pi(J0)=J^{c}0$$ ; then $$\pi$$ can be a permutation of $$P_{+}$$ only if $$c$$ is coprime to $$\overline{r}$$ .</p> <p>If $$k=1$$ then $$P_{+}=\{0,J0,\dots,J^{r}0\}$$ so $$\pi=\pi[c-1]$$ . Thus we can assume $$k\geq2$$ .</p> <p>Useful is the coefficient of $$\lambda$$ in the tensor product $$\Lambda_{1}\otimes\cdot\cdot\otimes\Lambda_{1}$$ ( $$\ell\,\mathrm{times})$$ : it is 0 unless $$t(\lambda)=\ell$$ , in which case the coefficient is $$\frac{\ell!}{h(\lambda)}$$ (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights $$\Lambda_{\overline{{r}}}$$ and $$\Lambda_{0}$$ , so e.g. $$^{\star}{\frac{k}{r}}\Lambda_{\overline{{r}}}^{\ \ \,}{}^{\ ,}$$ equals $$\mathrm{\Delta}^{\prime}0^{\circ}$$ when $$\overline{r}$$ divides $$k$$ . Here, $$h(\lambda)=\prod h(x)$$ is the product of the hook-lengths of the Young diagram corresponding to $$\lambda$$ . Equation (2.4) tells us that as long as $$t(\lambda)=\ell\leq k$$ , the number $$\frac{\ell!}{h(\lambda)}$$ will also be the coefficient of $$N_{\lambda}$$ in the fusion power $$(N_{\Lambda_{1}})^{\ell}$$ . Note that $$J0=k\Lambda_{1}$$ is the only simple-current appearing in the fusion product $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k$$ times). Thus the only nontrivial simple-current appearing in the fusion $$\pi\Lambda_{1}$$ × · · · × $$\pi\Lambda_{1}$$ will be $$J^{b k}J0$$ (0 will appear iff $$\overline{r}$$ divides $$k$$ ). Hence $$b k+1\equiv c$$ (mod $$\overline{r}$$ ) must be coprime to $$\overline{r}$$ . This is precisely the condition needed for $$\pi[b]$$ to be a simple-current automorphism.</p> <p>In other words, it suffices to consider $$\pi\Lambda_{1}=\Lambda_{1}$$ and hence $$\pi[J0]=J0$$ . We are done if $$r=1$$ , so assume $$r\geq2$$ . From the $$\Lambda_{1}$$ × $$\Lambda_{1}$$ fusion, we get that $$\pi\Lambda_{2}\in\{\Lambda_{2},2\Lambda_{1}\}$$ . Note that $$k\Lambda_{1}$$ occurs (with multiplicity 1) in the tensor and fusion product of $$2\Lambda_{1}$$ with $$k-2\;\Lambda_{1}\,{}^{\prime}\mathrm{s}$$ s, but that it doesn’t in the tensor (hence fusion) product of $$\Lambda_{2}$$ with $$k-2~\Lambda_{1}$$ ’s (recall that $$k\Lambda_{1}\succ(k-2)\Lambda_{1}+\Lambda_{2}$$ in the usual partial order on weights). Since $$\Lambda_{2}$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ does not contain $$J0$$ , $$(\pi\Lambda_{2})\boxtimes(\pi\Lambda_{1})\boxtimes\dots\boxtimes(\pi\Lambda_{1})$$ should also avoid $$\pi(J0)\,=\,J0$$ , and thus $$\pi\Lambda_{2}$$ cannot equal $$2\Lambda_{1}$$ .</p> <p>Thus we know $$\pi\Lambda_{2}=\Lambda_{2}$$ . The remaining $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ follow quickly from induction: if $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ for some $$2\leq\ell<r$$ , then the fusion $$\Lambda_{1}\boxtimes\Lambda_{\ell}$$ tells us $$\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$$ . But $$h(\Lambda_{1}+\Lambda_{\ell})=(\ell+1)!/\ell$$ and $$h(\Lambda_{\ell+1})=(\ell+1)!$$ , so $$\pi\Lambda_{\ell+1}=\Lambda_{\ell+1}$$ . Thus $$\pi$$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that $$\pi$$ must fix everything in $$P_{+}$$ .</p>	[{"type": "text", "coordinates": [71, 70, 542, 102], "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we\nwill sketch here the most interesting $$(A_{r}^{(1)})$$ and the most difficult $$(E_{8}^{(1)})$$ cases.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 103, 541, 192], "content": "Consider first $$A_{r,k}$$ . By choosing $$a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}$$ in (4.1), we get that either $$\\lambda=k\\Lambda_{\\ell}$$ for\nsome $$\\ell$$ , in which case $$\\lambda$$ is a simple-current and (for $$k\\neq1$$ ) $$D(\\lambda)<\\mathcal{D}(\\Lambda_{1})$$ , or $$\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})$$\nfor some $$\\ell$$ , with equality iff $$\\lambda\\in S\\Lambda_{\\ell}$$ . But then rank-level duality $$A_{r,k}\\leftrightarrow A_{k-1,r+1}$$ (defined\nas for $$C_{r,k}$$ , and which is exact for $$A_{r,k}$$ q-dimensions) and (4.1) with $$a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}$$ give\nus $$\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})$$ , with equality iff $$\\ell\\,=\\,1$$ or $$r$$ . Com bining these\nresults yields Proposition 4.1(a).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 193, 541, 288], "content": "For $$E_{8,k}$$ , run through each $$a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}$$ to reduce the proof to comparing\n$$\\mathcal{D}(\\Lambda_{1})$$ with $$\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$$ for $$i\\neq0$$ , or $$\\mathcal{D}(\\Lambda_{i})$$ for $$i\\neq0,1$$ (the argument in [18] unnecessarily\ncomplicated things by restricting to integral weights). Standard arguments (see [18] for\ndetails) quickly show that the q-dimension $$\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$$ monotonically increases with $$k$$ to $$\\infty$$ ,\nwhile $$\\mathcal{D}(\\Lambda_{i})$$ monotonically increases with $$k$$ to the Weyl dimension of $$\\Lambda_{i}$$ . The proof of\nProposition 4.1(e8) then reduces to a short computation.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [72, 301, 218, 316], "content": "4.2. The $$A$$ -series argument", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [70, 323, 541, 366], "content": "Recall that $$\\overline{r}\\,=\\,r\\,+\\,1$$ . Proposition 4.1(a) tells us that $$\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}$$ , for some\n$$a,b$$ . Hitting $$\\pi$$ with $$C^{a}$$ , we can assume without loss of generality that $$a\\;=\\;0$$ . Write\n$$\\pi(J0)=J^{c}0$$ ; then $$\\pi$$ can be a permutation of $$P_{+}$$ only if $$c$$ is coprime to $$\\overline{r}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [88, 365, 518, 380], "content": "If $$k=1$$ then $$P_{+}=\\{0,J0,\\dots,J^{r}0\\}$$ so $$\\pi=\\pi[c-1]$$ . Thus we can assume $$k\\geq2$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [69, 381, 541, 530], "content": "Useful is the coefficient of $$\\lambda$$ in the tensor product $$\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}$$ ( $$\\ell\\,\\mathrm{times})$$ : it is 0 unless\n$$t(\\lambda)=\\ell$$ , in which case the coefficient is $$\\frac{\\ell!}{h(\\lambda)}$$ (to get this, compare (3.1) above with [27,\np.114]) \u2014 we equate here the fundamental weights $$\\Lambda_{\\overline{{r}}}$$ and $$\\Lambda_{0}$$ , so e.g. $$^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}$$ equals $$\\mathrm{\\Delta}^{\\prime}0^{\\circ}$$ when\n$$\\overline{r}$$ divides $$k$$ . Here, $$h(\\lambda)=\\prod h(x)$$ is the product of the hook-lengths of the Young diagram\ncorresponding to $$\\lambda$$ . Equation (2.4) tells us that as long as $$t(\\lambda)=\\ell\\leq k$$ , the number $$\\frac{\\ell!}{h(\\lambda)}$$\nwill also be the coefficient of $$N_{\\lambda}$$ in the fusion power $$(N_{\\Lambda_{1}})^{\\ell}$$ . Note that $$J0=k\\Lambda_{1}$$ is the\nonly simple-current appearing in the fusion product $$\\Lambda_{1}$$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $$\\Lambda_{1}$$ ( $$k$$ times). Thus the\nonly nontrivial simple-current appearing in the fusion $$\\pi\\Lambda_{1}$$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $$\\pi\\Lambda_{1}$$ will be $$J^{b k}J0$$ (0\nwill appear iff $$\\overline{r}$$ divides $$k$$ ). Hence $$b k+1\\equiv c$$ (mod $$\\overline{r}$$ ) must be coprime to $$\\overline{r}$$ . This is\nprecisely the condition needed for $$\\pi[b]$$ to be a simple-current automorphism.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 531, 541, 631], "content": "In other words, it suffices to consider $$\\pi\\Lambda_{1}=\\Lambda_{1}$$ and hence $$\\pi[J0]=J0$$ . We are done if\n$$r=1$$ , so assume $$r\\geq2$$ . From the $$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{1}$$ fusion, we get that $$\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}$$ . Note that\n$$k\\Lambda_{1}$$ occurs (with multiplicity 1) in the tensor and fusion product of $$2\\Lambda_{1}$$ with $$k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}$$ s,\nbut that it doesn\u2019t in the tensor (hence fusion) product of $$\\Lambda_{2}$$ with $$k-2~\\Lambda_{1}$$ \u2019s (recall that\n$$k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}$$ in the usual partial order on weights). Since $$\\Lambda_{2}$$ \u00d7 $$\\Lambda_{1}$$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $$\\Lambda_{1}$$\ndoes not contain $$J0$$ , $$(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})$$ should also avoid $$\\pi(J0)\\,=\\,J0$$ , and\nthus $$\\pi\\Lambda_{2}$$ cannot equal $$2\\Lambda_{1}$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 632, 541, 703], "content": "Thus we know $$\\pi\\Lambda_{2}=\\Lambda_{2}$$ . The remaining $$\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$$ follow quickly from induction: if\n$$\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$$ for some $$2\\leq\\ell<r$$ , then the fusion $$\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}$$ tells us $$\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}$$ .\nBut $$h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell$$ and $$h(\\Lambda_{\\ell+1})=(\\ell+1)!$$ , so $$\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}$$ . Thus $$\\pi$$ fixes all\nfundamental weights, and since these comprise a fusion-generator (see the discussion at\nthe end of \u00a72.2) we know that $$\\pi$$ must fix everything in $$P_{+}$$ .", "block_type": "text", "index": 9}]	[{"type": "text", "coordinates": [95, 74, 541, 88], "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [68, 85, 264, 106], "content": "will sketch here the most interesting ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [265, 88, 296, 104], "content": "(A_{r}^{(1)})", "score": 0.88, "index": 3}, {"type": "text", "coordinates": [297, 85, 416, 106], "content": " and the most difficult ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [417, 88, 449, 104], "content": "(E_{8}^{(1)})", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [449, 85, 484, 106], "content": " cases.", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [94, 104, 167, 121], "content": "Consider first", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [168, 106, 190, 119], "content": "A_{r,k}", "score": 0.92, "index": 8}, {"type": "text", "coordinates": [190, 104, 263, 121], "content": ". By choosing ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [264, 106, 339, 119], "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "score": 0.91, "index": 10}, {"type": "text", "coordinates": [339, 104, 479, 121], "content": " in (4.1), we get that either ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [479, 106, 522, 117], "content": "\\lambda=k\\Lambda_{\\ell}", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [522, 104, 542, 121], "content": "for", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [70, 119, 100, 135], "content": "some ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [100, 121, 106, 129], "content": "\\ell", "score": 0.86, "index": 15}, {"type": "text", "coordinates": [106, 119, 182, 135], "content": ", in which case ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [182, 120, 190, 130], "content": "\\lambda", "score": 0.88, "index": 17}, {"type": "text", "coordinates": [190, 119, 335, 135], "content": " is a simple-current and (for ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [335, 120, 364, 132], "content": "k\\neq1", "score": 0.9, "index": 19}, {"type": "text", "coordinates": [364, 119, 371, 135], "content": ") ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [371, 120, 446, 133], "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "score": 0.93, "index": 21}, {"type": "text", "coordinates": [446, 119, 465, 135], "content": ", or ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [465, 120, 540, 133], "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [70, 132, 117, 150], "content": "for some ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [117, 135, 123, 144], "content": "\\ell", "score": 0.87, "index": 25}, {"type": "text", "coordinates": [123, 132, 213, 150], "content": ", with equality iff", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [214, 135, 256, 146], "content": "\\lambda\\in S\\Lambda_{\\ell}", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [257, 132, 407, 150], "content": ". But then rank-level duality ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [407, 135, 494, 147], "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [495, 132, 541, 150], "content": " (defined", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [71, 151, 104, 165], "content": "as for ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [104, 152, 126, 164], "content": "C_{r,k}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [126, 151, 251, 165], "content": ", and which is exact for ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [252, 152, 273, 164], "content": "A_{r,k}", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [273, 151, 432, 165], "content": " q-dimensions) and (4.1) with ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [432, 148, 515, 162], "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "score": 0.94, "index": 36}, {"type": "text", "coordinates": [515, 151, 540, 165], "content": " give", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [70, 165, 87, 182], "content": "us ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [87, 165, 282, 180], "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "score": 0.94, "index": 39}, {"type": "text", "coordinates": [282, 165, 380, 182], "content": ", with equality iff", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [380, 168, 412, 177], "content": "\\ell\\,=\\,1", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [412, 165, 432, 182], "content": " or ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [433, 171, 439, 177], "content": "r", "score": 0.86, "index": 43}, {"type": "text", "coordinates": [439, 165, 541, 182], "content": ". Com bining these", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [71, 181, 243, 196], "content": "results yields Proposition 4.1(a).", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [93, 194, 117, 212], "content": "For ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [117, 197, 140, 209], "content": "E_{8,k}", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [140, 194, 244, 212], "content": ", run through each ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [244, 196, 357, 211], "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "score": 0.94, "index": 49}, {"type": "text", "coordinates": [358, 194, 541, 212], "content": " to reduce the proof to comparing", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [71, 213, 104, 226], "content": "\\mathcal{D}(\\Lambda_{1})", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [104, 209, 135, 232], "content": " with ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [136, 212, 181, 230], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "score": 0.95, "index": 53}, {"type": "text", "coordinates": [182, 209, 204, 232], "content": " for ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [204, 214, 232, 225], "content": "i\\neq0", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [233, 209, 255, 232], "content": ", or ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [255, 213, 286, 226], "content": "\\mathcal{D}(\\Lambda_{i})", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [287, 209, 309, 232], "content": " for ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [309, 214, 349, 225], "content": "i\\neq0,1", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [349, 209, 543, 232], "content": " (the argument in [18] unnecessarily", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [71, 228, 540, 244], "content": "complicated things by restricting to integral weights). Standard arguments (see [18] for", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [69, 241, 296, 262], "content": "details) quickly show that the q-dimension ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [297, 243, 342, 261], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "score": 0.95, "index": 63}, {"type": "text", "coordinates": [342, 241, 499, 262], "content": " monotonically increases with ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [499, 245, 506, 254], "content": "k", "score": 0.89, "index": 65}, {"type": "text", "coordinates": [506, 241, 524, 262], "content": " to ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [524, 249, 536, 254], "content": "\\infty", "score": 0.84, "index": 67}, {"type": "text", "coordinates": [537, 241, 543, 262], "content": ",", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [72, 261, 103, 275], "content": "while ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [104, 262, 135, 274], "content": "\\mathcal{D}(\\Lambda_{i})", "score": 0.94, "index": 70}, {"type": "text", "coordinates": [135, 261, 296, 275], "content": " monotonically increases with ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [297, 263, 304, 271], "content": "k", "score": 0.89, "index": 72}, {"type": "text", "coordinates": [304, 261, 448, 275], "content": " to the Weyl dimension of ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [448, 263, 461, 273], "content": "\\Lambda_{i}", "score": 0.92, "index": 74}, {"type": "text", "coordinates": [461, 261, 542, 275], "content": ". The proof of", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [70, 276, 371, 290], "content": "Proposition 4.1(e8) then reduces to a short computation.", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [71, 304, 120, 316], "content": "4.2. The ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [121, 306, 130, 314], "content": "A", "score": 0.33, "index": 78}, {"type": "text", "coordinates": [130, 304, 219, 316], "content": "-series argument", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [93, 325, 159, 339], "content": "Recall that ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [159, 327, 215, 337], "content": "\\overline{r}\\,=\\,r\\,+\\,1", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [216, 325, 401, 339], "content": ". Proposition 4.1(a) tells us that ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [401, 325, 484, 338], "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "score": 0.94, "index": 83}, {"type": "text", "coordinates": [484, 325, 541, 339], "content": ", for some", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [71, 342, 88, 353], "content": "a,b", "score": 0.91, "index": 85}, {"type": "text", "coordinates": [89, 339, 144, 354], "content": ". Hitting ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [144, 345, 151, 350], "content": "\\pi", "score": 0.84, "index": 87}, {"type": "text", "coordinates": [152, 339, 185, 354], "content": " with ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [185, 342, 200, 350], "content": "C^{a}", "score": 0.9, "index": 89}, {"type": "text", "coordinates": [201, 339, 464, 354], "content": ", we can assume without loss of generality that ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [464, 342, 497, 351], "content": "a\\;=\\;0", "score": 0.91, "index": 91}, {"type": "text", "coordinates": [497, 339, 542, 354], "content": ". Write", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [71, 355, 137, 368], "content": "\\pi(J0)=J^{c}0", "score": 0.92, "index": 93}, {"type": "text", "coordinates": [137, 353, 170, 369], "content": "; then ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [171, 359, 178, 365], "content": "\\pi", "score": 0.87, "index": 95}, {"type": "text", "coordinates": [178, 353, 312, 369], "content": " can be a permutation of ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [312, 356, 327, 367], "content": "P_{+}", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [328, 353, 369, 369], "content": " only if ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [369, 359, 375, 365], "content": "c", "score": 0.88, "index": 99}, {"type": "text", "coordinates": [375, 353, 449, 369], "content": " is coprime to ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [450, 357, 456, 365], "content": "\\overline{r}", "score": 0.89, "index": 101}, {"type": "text", "coordinates": [456, 353, 461, 369], "content": ".", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [93, 366, 106, 384], "content": "If ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [107, 370, 136, 379], "content": "k=1", "score": 0.92, "index": 104}, {"type": "text", "coordinates": [136, 366, 166, 384], "content": " then ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [167, 370, 281, 382], "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [281, 366, 299, 384], "content": " so ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [299, 370, 363, 382], "content": "\\pi=\\pi[c-1]", "score": 0.94, "index": 108}, {"type": "text", "coordinates": [363, 366, 482, 384], "content": ". Thus we can assume ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [482, 370, 511, 381], "content": "k\\geq2", "score": 0.95, "index": 110}, {"type": "text", "coordinates": [511, 366, 516, 384], "content": ".", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [93, 381, 230, 398], "content": "Useful is the coefficient of ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [231, 384, 238, 393], "content": "\\lambda", "score": 0.89, "index": 113}, {"type": "text", "coordinates": [238, 381, 353, 398], "content": " in the tensor product ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [353, 384, 419, 395], "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [419, 381, 426, 398], "content": " (", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [426, 384, 468, 396], "content": "\\ell\\,\\mathrm{times})", "score": 0.29, "index": 117}, {"type": "text", "coordinates": [468, 381, 541, 398], "content": ": it is 0 unless", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [71, 398, 115, 411], "content": "t(\\lambda)=\\ell", "score": 0.93, "index": 119}, {"type": "text", "coordinates": [115, 393, 288, 419], "content": ", in which case the coefficient is", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [288, 396, 309, 414], "content": "\\frac{\\ell!}{h(\\lambda)}", "score": 0.94, "index": 121}, {"type": "text", "coordinates": [309, 393, 543, 419], "content": " (to get this, compare (3.1) above with [27,", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [70, 414, 334, 430], "content": "p.114]) \u2014 we equate here the fundamental weights ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [335, 416, 349, 427], "content": "\\Lambda_{\\overline{{r}}}", "score": 0.91, "index": 124}, {"type": "text", "coordinates": [349, 414, 373, 430], "content": " and", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [374, 417, 388, 428], "content": "\\Lambda_{0}", "score": 0.9, "index": 126}, {"type": "text", "coordinates": [388, 414, 430, 430], "content": ", so e.g. ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [430, 414, 458, 430], "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "score": 0.91, "index": 128}, {"type": "text", "coordinates": [458, 414, 498, 430], "content": " equals ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [498, 416, 509, 426], "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "score": 0.39, "index": 130}, {"type": "text", "coordinates": [509, 414, 541, 430], "content": " when", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [71, 432, 77, 440], "content": "\\overline{r}", "score": 0.86, "index": 132}, {"type": "text", "coordinates": [78, 429, 120, 445], "content": " divides ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [120, 431, 127, 440], "content": "k", "score": 0.89, "index": 134}, {"type": "text", "coordinates": [128, 429, 167, 445], "content": ". Here, ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [167, 430, 242, 443], "content": "h(\\lambda)=\\prod h(x)", "score": 0.95, "index": 136}, {"type": "text", "coordinates": [243, 429, 541, 445], "content": " is the product of the hook-lengths of the Young diagram", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [70, 443, 162, 459], "content": "corresponding to ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [163, 446, 170, 454], "content": "\\lambda", "score": 0.88, "index": 139}, {"type": "text", "coordinates": [171, 443, 381, 459], "content": ". Equation (2.4) tells us that as long as ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [381, 444, 446, 457], "content": "t(\\lambda)=\\ell\\leq k", "score": 0.91, "index": 141}, {"type": "text", "coordinates": [447, 443, 517, 459], "content": ", the number", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [518, 443, 539, 460], "content": "\\frac{\\ell!}{h(\\lambda)}", "score": 0.93, "index": 143}, {"type": "text", "coordinates": [69, 460, 226, 476], "content": "will also be the coefficient of ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [227, 463, 243, 474], "content": "N_{\\lambda}", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [243, 460, 353, 476], "content": " in the fusion power ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [353, 461, 389, 475], "content": "(N_{\\Lambda_{1}})^{\\ell}", "score": 0.92, "index": 147}, {"type": "text", "coordinates": [389, 460, 454, 476], "content": ". Note that ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [454, 463, 506, 474], "content": "J0=k\\Lambda_{1}", "score": 0.92, "index": 149}, {"type": "text", "coordinates": [506, 460, 542, 476], "content": " is the", "score": 1.0, "index": 150}, {"type": "text", "coordinates": [70, 475, 348, 491], "content": "only simple-current appearing in the fusion product ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [348, 476, 362, 488], "content": "\\Lambda_{1}", "score": 0.73, "index": 152}, {"type": "text", "coordinates": [362, 475, 415, 491], "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [415, 475, 430, 488], "content": "\\Lambda_{1}", "score": 0.84, "index": 154}, {"type": "text", "coordinates": [430, 475, 438, 491], "content": " (", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [439, 476, 446, 487], "content": "k", "score": 0.67, "index": 156}, {"type": "text", "coordinates": [446, 475, 542, 491], "content": " times). Thus the", "score": 1.0, "index": 157}, {"type": "text", "coordinates": [70, 489, 354, 505], "content": "only nontrivial simple-current appearing in the fusion ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [354, 489, 376, 502], "content": "\\pi\\Lambda_{1}", "score": 0.81, "index": 159}, {"type": "text", "coordinates": [376, 489, 429, 505], "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [429, 489, 451, 502], "content": "\\pi\\Lambda_{1}", "score": 0.87, "index": 161}, {"type": "text", "coordinates": [451, 489, 492, 505], "content": " will be", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [493, 489, 525, 501], "content": "J^{b k}J0", "score": 0.83, "index": 163}, {"type": "text", "coordinates": [526, 489, 542, 505], "content": " (0", "score": 1.0, "index": 164}, {"type": "text", "coordinates": [72, 505, 150, 518], "content": "will appear iff", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [151, 507, 157, 515], "content": "\\overline{r}", "score": 0.86, "index": 166}, {"type": "text", "coordinates": [158, 505, 203, 518], "content": " divides ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [203, 506, 210, 515], "content": "k", "score": 0.82, "index": 168}, {"type": "text", "coordinates": [210, 505, 262, 518], "content": "). Hence ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [263, 506, 322, 516], "content": "b k+1\\equiv c", "score": 0.89, "index": 170}, {"type": "text", "coordinates": [323, 505, 358, 518], "content": " (mod ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [359, 506, 366, 515], "content": "\\overline{r}", "score": 0.59, "index": 172}, {"type": "text", "coordinates": [366, 505, 484, 518], "content": ") must be coprime to ", "score": 1.0, "index": 173}, {"type": "inline_equation", "coordinates": [484, 506, 491, 515], "content": "\\overline{r}", "score": 0.76, "index": 174}, {"type": "text", "coordinates": [492, 505, 541, 518], "content": ". This is", "score": 1.0, "index": 175}, {"type": "text", "coordinates": [72, 520, 251, 533], "content": "precisely the condition needed for ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [251, 519, 271, 532], "content": "\\pi[b]", "score": 0.92, "index": 177}, {"type": "text", "coordinates": [271, 520, 472, 533], "content": " to be a simple-current automorphism.", "score": 1.0, "index": 178}, {"type": "text", "coordinates": [94, 533, 289, 547], "content": "In other words, it suffices to consider ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [289, 533, 340, 545], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "score": 0.92, "index": 180}, {"type": "text", "coordinates": [340, 533, 398, 547], "content": " and hence ", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [398, 532, 457, 546], "content": "\\pi[J0]=J0", "score": 0.92, "index": 182}, {"type": "text", "coordinates": [457, 533, 543, 547], "content": ". We are done if", "score": 1.0, "index": 183}, {"type": "inline_equation", "coordinates": [71, 550, 99, 558], "content": "r=1", "score": 0.89, "index": 184}, {"type": "text", "coordinates": [100, 547, 160, 563], "content": ", so assume ", "score": 1.0, "index": 185}, {"type": "inline_equation", "coordinates": [160, 549, 188, 560], "content": "r\\geq2", "score": 0.89, "index": 186}, {"type": "text", "coordinates": [189, 547, 246, 563], "content": ". From the ", "score": 1.0, "index": 187}, {"type": "inline_equation", "coordinates": [246, 547, 261, 560], "content": "\\Lambda_{1}", "score": 0.78, "index": 188}, {"type": "text", "coordinates": [261, 547, 277, 563], "content": " \u00d7", "score": 1.0, "index": 189}, {"type": "inline_equation", "coordinates": [278, 547, 293, 560], "content": "\\Lambda_{1}", "score": 0.77, "index": 190}, {"type": "text", "coordinates": [293, 547, 394, 563], "content": " fusion, we get that ", "score": 1.0, "index": 191}, {"type": "inline_equation", "coordinates": [394, 547, 481, 561], "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "score": 0.91, "index": 192}, {"type": "text", "coordinates": [482, 547, 541, 563], "content": ". Note that", "score": 1.0, "index": 193}, {"type": "inline_equation", "coordinates": [71, 563, 91, 574], "content": "k\\Lambda_{1}", "score": 0.91, "index": 194}, {"type": "text", "coordinates": [92, 561, 431, 576], "content": " occurs (with multiplicity 1) in the tensor and fusion product of ", "score": 1.0, "index": 195}, {"type": "inline_equation", "coordinates": [431, 561, 452, 574], "content": "2\\Lambda_{1}", "score": 0.9, "index": 196}, {"type": "text", "coordinates": [452, 561, 482, 576], "content": " with ", "score": 1.0, "index": 197}, {"type": "inline_equation", "coordinates": [482, 561, 534, 574], "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "score": 0.69, "index": 198}, {"type": "text", "coordinates": [534, 561, 540, 576], "content": "s,", "score": 1.0, "index": 199}, {"type": "text", "coordinates": [70, 576, 378, 590], "content": "but that it doesn\u2019t in the tensor (hence fusion) product of ", "score": 1.0, "index": 200}, {"type": "inline_equation", "coordinates": [379, 576, 393, 588], "content": "\\Lambda_{2}", "score": 0.89, "index": 201}, {"type": "text", "coordinates": [393, 576, 423, 590], "content": " with ", "score": 1.0, "index": 202}, {"type": "inline_equation", "coordinates": [423, 575, 469, 588], "content": "k-2~\\Lambda_{1}", "score": 0.79, "index": 203}, {"type": "text", "coordinates": [470, 576, 541, 590], "content": "\u2019s (recall that", "score": 1.0, "index": 204}, {"type": "inline_equation", "coordinates": [71, 591, 186, 604], "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "score": 0.92, "index": 205}, {"type": "text", "coordinates": [187, 590, 425, 606], "content": " in the usual partial order on weights). Since ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [425, 590, 440, 603], "content": "\\Lambda_{2}", "score": 0.72, "index": 207}, {"type": "text", "coordinates": [440, 590, 456, 606], "content": " \u00d7", "score": 1.0, "index": 208}, {"type": "inline_equation", "coordinates": [457, 590, 472, 603], "content": "\\Lambda_{1}", "score": 0.74, "index": 209}, {"type": "text", "coordinates": [472, 590, 524, 606], "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "score": 1.0, "index": 210}, {"type": "inline_equation", "coordinates": [524, 589, 540, 603], "content": "\\Lambda_{1}", "score": 0.84, "index": 211}, {"type": "text", "coordinates": [69, 604, 163, 621], "content": "does not contain ", "score": 1.0, "index": 212}, {"type": "inline_equation", "coordinates": [164, 606, 178, 616], "content": "J0", "score": 0.8, "index": 213}, {"type": "text", "coordinates": [178, 604, 185, 621], "content": ", ", "score": 1.0, "index": 214}, {"type": "inline_equation", "coordinates": [185, 603, 348, 618], "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "score": 0.25, "index": 215}, {"type": "text", "coordinates": [349, 604, 449, 621], "content": " should also avoid ", "score": 1.0, "index": 216}, {"type": "inline_equation", "coordinates": [449, 604, 513, 618], "content": "\\pi(J0)\\,=\\,J0", "score": 0.93, "index": 217}, {"type": "text", "coordinates": [513, 604, 542, 621], "content": ", and", "score": 1.0, "index": 218}, {"type": "text", "coordinates": [70, 618, 97, 634], "content": "thus ", "score": 1.0, "index": 219}, {"type": "inline_equation", "coordinates": [97, 621, 118, 631], "content": "\\pi\\Lambda_{2}", "score": 0.92, "index": 220}, {"type": "text", "coordinates": [119, 618, 192, 634], "content": " cannot equal ", "score": 1.0, "index": 221}, {"type": "inline_equation", "coordinates": [193, 619, 213, 632], "content": "2\\Lambda_{1}", "score": 0.9, "index": 222}, {"type": "text", "coordinates": [213, 618, 218, 634], "content": ".", "score": 1.0, "index": 223}, {"type": "text", "coordinates": [94, 632, 174, 649], "content": "Thus we know ", "score": 1.0, "index": 224}, {"type": "inline_equation", "coordinates": [174, 633, 226, 646], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "score": 0.91, "index": 225}, {"type": "text", "coordinates": [226, 632, 315, 649], "content": ". The remaining ", "score": 1.0, "index": 226}, {"type": "inline_equation", "coordinates": [316, 633, 366, 646], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "score": 0.92, "index": 227}, {"type": "text", "coordinates": [366, 632, 543, 649], "content": "follow quickly from induction: if", "score": 1.0, "index": 228}, {"type": "inline_equation", "coordinates": [71, 649, 121, 660], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "score": 0.92, "index": 229}, {"type": "text", "coordinates": [121, 647, 173, 663], "content": "for some ", "score": 1.0, "index": 230}, {"type": "inline_equation", "coordinates": [173, 648, 224, 660], "content": "2\\leq\\ell<r", "score": 0.87, "index": 231}, {"type": "text", "coordinates": [225, 647, 314, 663], "content": ", then the fusion ", "score": 1.0, "index": 232}, {"type": "inline_equation", "coordinates": [314, 647, 360, 660], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "score": 0.44, "index": 233}, {"type": "text", "coordinates": [360, 647, 404, 663], "content": "tells us ", "score": 1.0, "index": 234}, {"type": "inline_equation", "coordinates": [405, 648, 537, 662], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "score": 0.93, "index": 235}, {"type": "text", "coordinates": [537, 647, 541, 663], "content": ".", "score": 1.0, "index": 236}, {"type": "text", "coordinates": [70, 661, 95, 678], "content": "But ", "score": 1.0, "index": 237}, {"type": "inline_equation", "coordinates": [96, 662, 222, 675], "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "score": 0.92, "index": 238}, {"type": "text", "coordinates": [222, 661, 249, 678], "content": "and ", "score": 1.0, "index": 239}, {"type": "inline_equation", "coordinates": [250, 662, 347, 676], "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "score": 0.92, "index": 240}, {"type": "text", "coordinates": [348, 661, 370, 678], "content": ", so ", "score": 1.0, "index": 241}, {"type": "inline_equation", "coordinates": [371, 663, 446, 675], "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "score": 0.92, "index": 242}, {"type": "text", "coordinates": [446, 661, 487, 678], "content": ". Thus ", "score": 1.0, "index": 243}, {"type": "inline_equation", "coordinates": [487, 664, 495, 673], "content": "\\pi", "score": 0.72, "index": 244}, {"type": "text", "coordinates": [495, 661, 542, 678], "content": " fixes all", "score": 1.0, "index": 245}, {"type": "text", "coordinates": [70, 676, 542, 691], "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "score": 1.0, "index": 246}, {"type": "text", "coordinates": [70, 689, 231, 707], "content": "the end of \u00a72.2) we know that ", "score": 1.0, "index": 247}, {"type": "inline_equation", "coordinates": [232, 695, 240, 701], "content": "\\pi", "score": 0.67, "index": 248}, {"type": "text", "coordinates": [240, 689, 362, 707], "content": " must fix everything in ", "score": 1.0, "index": 249}, {"type": "inline_equation", "coordinates": [362, 692, 378, 704], "content": "P_{+}", "score": 0.9, "index": 250}, {"type": "text", "coordinates": [379, 689, 384, 707], "content": ".", "score": 1.0, "index": 251}]	[]	[{"type": "inline", "coordinates": [265, 88, 296, 104], "content": "(A_{r}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [417, 88, 449, 104], "content": "(E_{8}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [168, 106, 190, 119], "content": "A_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [264, 106, 339, 119], "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "caption": ""}, {"type": "inline", "coordinates": [479, 106, 522, 117], "content": "\\lambda=k\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [100, 121, 106, 129], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [182, 120, 190, 130], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [335, 120, 364, 132], "content": "k\\neq1", "caption": ""}, {"type": "inline", "coordinates": [371, 120, 446, 133], "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [465, 120, 540, 133], "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "caption": ""}, {"type": "inline", "coordinates": [117, 135, 123, 144], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [214, 135, 256, 146], "content": "\\lambda\\in S\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [407, 135, 494, 147], "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "caption": ""}, {"type": "inline", "coordinates": [104, 152, 126, 164], "content": "C_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [252, 152, 273, 164], "content": "A_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [432, 148, 515, 162], "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "caption": ""}, {"type": "inline", "coordinates": [87, 165, 282, 180], "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [380, 168, 412, 177], "content": "\\ell\\,=\\,1", "caption": ""}, {"type": "inline", "coordinates": [433, 171, 439, 177], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [117, 197, 140, 209], "content": "E_{8,k}", "caption": ""}, {"type": "inline", "coordinates": [244, 196, 357, 211], "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "caption": ""}, {"type": "inline", "coordinates": [71, 213, 104, 226], "content": "\\mathcal{D}(\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [136, 212, 181, 230], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "caption": ""}, {"type": "inline", "coordinates": [204, 214, 232, 225], "content": "i\\neq0", "caption": ""}, {"type": "inline", "coordinates": [255, 213, 286, 226], "content": "\\mathcal{D}(\\Lambda_{i})", "caption": ""}, {"type": "inline", "coordinates": [309, 214, 349, 225], "content": "i\\neq0,1", "caption": ""}, {"type": "inline", "coordinates": [297, 243, 342, 261], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "caption": ""}, {"type": "inline", "coordinates": [499, 245, 506, 254], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [524, 249, 536, 254], "content": "\\infty", "caption": ""}, {"type": "inline", "coordinates": [104, 262, 135, 274], "content": "\\mathcal{D}(\\Lambda_{i})", "caption": ""}, {"type": "inline", "coordinates": [297, 263, 304, 271], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [448, 263, 461, 273], "content": "\\Lambda_{i}", "caption": ""}, {"type": "inline", "coordinates": [121, 306, 130, 314], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [159, 327, 215, 337], "content": "\\overline{r}\\,=\\,r\\,+\\,1", "caption": ""}, {"type": "inline", "coordinates": [401, 325, 484, 338], "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [71, 342, 88, 353], "content": "a,b", "caption": ""}, {"type": "inline", "coordinates": [144, 345, 151, 350], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [185, 342, 200, 350], "content": "C^{a}", "caption": ""}, {"type": "inline", "coordinates": [464, 342, 497, 351], "content": "a\\;=\\;0", "caption": ""}, {"type": "inline", "coordinates": [71, 355, 137, 368], "content": "\\pi(J0)=J^{c}0", "caption": ""}, {"type": "inline", "coordinates": [171, 359, 178, 365], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [312, 356, 327, 367], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [369, 359, 375, 365], "content": "c", "caption": ""}, {"type": "inline", "coordinates": [450, 357, 456, 365], "content": "\\overline{r}", "caption": ""}, {"type": "inline", "coordinates": [107, 370, 136, 379], "content": "k=1", "caption": ""}, {"type": "inline", "coordinates": [167, 370, 281, 382], "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "caption": ""}, {"type": "inline", "coordinates": [299, 370, 363, 382], "content": "\\pi=\\pi[c-1]", "caption": ""}, {"type": "inline", "coordinates": [482, 370, 511, 381], "content": "k\\geq2", "caption": ""}, {"type": "inline", "coordinates": [231, 384, 238, 393], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [353, 384, 419, 395], "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [426, 384, 468, 396], "content": "\\ell\\,\\mathrm{times})", "caption": ""}, {"type": "inline", "coordinates": [71, 398, 115, 411], "content": "t(\\lambda)=\\ell", "caption": ""}, {"type": "inline", "coordinates": [288, 396, 309, 414], "content": "\\frac{\\ell!}{h(\\lambda)}", "caption": ""}, {"type": "inline", "coordinates": [335, 416, 349, 427], "content": "\\Lambda_{\\overline{{r}}}", "caption": ""}, {"type": "inline", "coordinates": [374, 417, 388, 428], "content": "\\Lambda_{0}", "caption": ""}, {"type": "inline", "coordinates": [430, 414, 458, 430], "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "caption": ""}, {"type": "inline", "coordinates": [498, 416, 509, 426], "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "caption": ""}, {"type": "inline", "coordinates": [71, 432, 77, 440], "content": "\\overline{r}", "caption": ""}, {"type": "inline", "coordinates": [120, 431, 127, 440], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [167, 430, 242, 443], "content": "h(\\lambda)=\\prod h(x)", "caption": ""}, {"type": "inline", "coordinates": [163, 446, 170, 454], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [381, 444, 446, 457], "content": "t(\\lambda)=\\ell\\leq k", "caption": ""}, {"type": "inline", "coordinates": [518, 443, 539, 460], "content": "\\frac{\\ell!}{h(\\lambda)}", "caption": ""}, {"type": "inline", "coordinates": [227, 463, 243, 474], "content": "N_{\\lambda}", "caption": ""}, {"type": "inline", "coordinates": [353, 461, 389, 475], "content": "(N_{\\Lambda_{1}})^{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [454, 463, 506, 474], "content": "J0=k\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [348, 476, 362, 488], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [415, 475, 430, 488], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [439, 476, 446, 487], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [354, 489, 376, 502], "content": "\\pi\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [429, 489, 451, 502], "content": "\\pi\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [493, 489, 525, 501], "content": "J^{b k}J0", "caption": ""}, {"type": "inline", "coordinates": [151, 507, 157, 515], "content": "\\overline{r}", "caption": ""}, {"type": "inline", "coordinates": [203, 506, 210, 515], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [263, 506, 322, 516], "content": "b k+1\\equiv c", "caption": ""}, {"type": "inline", "coordinates": [359, 506, 366, 515], "content": "\\overline{r}", "caption": ""}, {"type": "inline", "coordinates": [484, 506, 491, 515], "content": "\\overline{r}", "caption": ""}, {"type": "inline", "coordinates": [251, 519, 271, 532], "content": "\\pi[b]", "caption": ""}, {"type": "inline", "coordinates": [289, 533, 340, 545], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [398, 532, 457, 546], "content": "\\pi[J0]=J0", "caption": ""}, {"type": "inline", "coordinates": [71, 550, 99, 558], "content": "r=1", "caption": ""}, {"type": "inline", "coordinates": [160, 549, 188, 560], "content": "r\\geq2", "caption": ""}, {"type": "inline", "coordinates": [246, 547, 261, 560], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [278, 547, 293, 560], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [394, 547, 481, 561], "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "caption": ""}, {"type": "inline", "coordinates": [71, 563, 91, 574], "content": "k\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [431, 561, 452, 574], "content": "2\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [482, 561, 534, 574], "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "caption": ""}, {"type": "inline", "coordinates": [379, 576, 393, 588], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [423, 575, 469, 588], "content": "k-2~\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [71, 591, 186, 604], "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [425, 590, 440, 603], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [457, 590, 472, 603], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [524, 589, 540, 603], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [164, 606, 178, 616], "content": "J0", "caption": ""}, {"type": "inline", "coordinates": [185, 603, 348, 618], "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [449, 604, 513, 618], "content": "\\pi(J0)\\,=\\,J0", "caption": ""}, {"type": "inline", "coordinates": [97, 621, 118, 631], "content": "\\pi\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [193, 619, 213, 632], "content": "2\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [174, 633, 226, 646], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [316, 633, 366, 646], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [71, 649, 121, 660], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [173, 648, 224, 660], "content": "2\\leq\\ell<r", "caption": ""}, {"type": "inline", "coordinates": [314, 647, 360, 660], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [405, 648, 537, 662], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "caption": ""}, {"type": "inline", "coordinates": [96, 662, 222, 675], "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "caption": ""}, {"type": "inline", "coordinates": [250, 662, 347, 676], "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "caption": ""}, {"type": "inline", "coordinates": [371, 663, 446, 675], "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "caption": ""}, {"type": "inline", "coordinates": [487, 664, 495, 673], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [232, 695, 240, 701], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [362, 692, 378, 704], "content": "P_{+}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $(A_{r}^{(1)})$ and the most difficult $(E_{8}^{(1)})$ cases. ", "page_idx": 16}, {"type": "text", "text": "Consider first $A_{r,k}$ . By choosing $a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}$ in (4.1), we get that either $\\lambda=k\\Lambda_{\\ell}$ for some $\\ell$ , in which case $\\lambda$ is a simple-current and (for $k\\neq1$ ) $D(\\lambda)<\\mathcal{D}(\\Lambda_{1})$ , or $\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})$ for some $\\ell$ , with equality iff $\\lambda\\in S\\Lambda_{\\ell}$ . But then rank-level duality $A_{r,k}\\leftrightarrow A_{k-1,r+1}$ (defined as for $C_{r,k}$ , and which is exact for $A_{r,k}$ q-dimensions) and (4.1) with $a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}$ give us $\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})$ , with equality iff $\\ell\\,=\\,1$ or $r$ . Com bining these results yields Proposition 4.1(a). ", "page_idx": 16}, {"type": "text", "text": "For $E_{8,k}$ , run through each $a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}$ to reduce the proof to comparing $\\mathcal{D}(\\Lambda_{1})$ with $\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$ for $i\\neq0$ , or $\\mathcal{D}(\\Lambda_{i})$ for $i\\neq0,1$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$ monotonically increases with $k$ to $\\infty$ , while $\\mathcal{D}(\\Lambda_{i})$ monotonically increases with $k$ to the Weyl dimension of $\\Lambda_{i}$ . The proof of Proposition 4.1(e8) then reduces to a short computation. ", "page_idx": 16}, {"type": "text", "text": "4.2. The $A$ -series argument ", "text_level": 1, "page_idx": 16}, {"type": "text", "text": "Recall that $\\overline{r}\\,=\\,r\\,+\\,1$ . Proposition 4.1(a) tells us that $\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}$ , for some $a,b$ . Hitting $\\pi$ with $C^{a}$ , we can assume without loss of generality that $a\\;=\\;0$ . Write $\\pi(J0)=J^{c}0$ ; then $\\pi$ can be a permutation of $P_{+}$ only if $c$ is coprime to $\\overline{r}$ . ", "page_idx": 16}, {"type": "text", "text": "If $k=1$ then $P_{+}=\\{0,J0,\\dots,J^{r}0\\}$ so $\\pi=\\pi[c-1]$ . Thus we can assume $k\\geq2$ . ", "page_idx": 16}, {"type": "text", "text": "Useful is the coefficient of $\\lambda$ in the tensor product $\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}$ ( $\\ell\\,\\mathrm{times})$ : it is 0 unless $t(\\lambda)=\\ell$ , in which case the coefficient is $\\frac{\\ell!}{h(\\lambda)}$ (to get this, compare (3.1) above with [27, p.114]) \u2014 we equate here the fundamental weights $\\Lambda_{\\overline{{r}}}$ and $\\Lambda_{0}$ , so e.g. $^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}$ equals $\\mathrm{\\Delta}^{\\prime}0^{\\circ}$ when $\\overline{r}$ divides $k$ . Here, $h(\\lambda)=\\prod h(x)$ is the product of the hook-lengths of the Young diagram corresponding to $\\lambda$ . Equation (2.4) tells us that as long as $t(\\lambda)=\\ell\\leq k$ , the number $\\frac{\\ell!}{h(\\lambda)}$ will also be the coefficient of $N_{\\lambda}$ in the fusion power $(N_{\\Lambda_{1}})^{\\ell}$ . Note that $J0=k\\Lambda_{1}$ is the only simple-current appearing in the fusion product $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k$ times). Thus the only nontrivial simple-current appearing in the fusion $\\pi\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\pi\\Lambda_{1}$ will be $J^{b k}J0$ (0 will appear iff $\\overline{r}$ divides $k$ ). Hence $b k+1\\equiv c$ (mod $\\overline{r}$ ) must be coprime to $\\overline{r}$ . This is precisely the condition needed for $\\pi[b]$ to be a simple-current automorphism. ", "page_idx": 16}, {"type": "text", "text": "In other words, it suffices to consider $\\pi\\Lambda_{1}=\\Lambda_{1}$ and hence $\\pi[J0]=J0$ . We are done if $r=1$ , so assume $r\\geq2$ . From the $\\Lambda_{1}$ \u00d7 $\\Lambda_{1}$ fusion, we get that $\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}$ . Note that $k\\Lambda_{1}$ occurs (with multiplicity 1) in the tensor and fusion product of $2\\Lambda_{1}$ with $k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}$ s, but that it doesn\u2019t in the tensor (hence fusion) product of $\\Lambda_{2}$ with $k-2~\\Lambda_{1}$ \u2019s (recall that $k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}$ in the usual partial order on weights). Since $\\Lambda_{2}$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ does not contain $J0$ , $(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})$ should also avoid $\\pi(J0)\\,=\\,J0$ , and thus $\\pi\\Lambda_{2}$ cannot equal $2\\Lambda_{1}$ . ", "page_idx": 16}, {"type": "text", "text": "Thus we know $\\pi\\Lambda_{2}=\\Lambda_{2}$ . The remaining $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ follow quickly from induction: if $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ for some $2\\leq\\ell<r$ , then the fusion $\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}$ tells us $\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}$ . But $h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell$ and $h(\\Lambda_{\\ell+1})=(\\ell+1)!$ , so $\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}$ . Thus $\\pi$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of \u00a72.2) we know that $\\pi$ must fix everything in $P_{+}$ . 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Standard arguments (see [18] for", "type": "text"}], "index": 10}, {"bbox": [69, 241, 543, 262], "spans": [{"bbox": [69, 241, 296, 262], "score": 1.0, "content": "details) quickly show that the q-dimension ", "type": "text"}, {"bbox": [297, 243, 342, 261], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [342, 241, 499, 262], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [499, 245, 506, 254], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [506, 241, 524, 262], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [524, 249, 536, 254], "score": 0.84, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 12}, {"bbox": [537, 241, 543, 262], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [72, 261, 542, 275], "spans": [{"bbox": [72, 261, 103, 275], "score": 1.0, "content": "while ", "type": "text"}, {"bbox": [104, 262, 135, 274], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [135, 261, 296, 275], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [297, 263, 304, 271], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [304, 261, 448, 275], "score": 1.0, "content": " to the Weyl dimension of ", "type": "text"}, {"bbox": [448, 263, 461, 273], "score": 0.92, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [461, 261, 542, 275], "score": 1.0, "content": ". The proof of", "type": "text"}], "index": 12}, {"bbox": [70, 276, 371, 290], "spans": [{"bbox": [70, 276, 371, 290], "score": 1.0, "content": "Proposition 4.1(e8) then reduces to a short computation.", "type": "text"}], "index": 13}], "index": 10.5}, {"type": "title", "bbox": [72, 301, 218, 316], "lines": [{"bbox": [71, 304, 219, 316], "spans": [{"bbox": [71, 304, 120, 316], "score": 1.0, "content": "4.2. The ", "type": "text"}, {"bbox": [121, 306, 130, 314], "score": 0.33, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [130, 304, 219, 316], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 323, 541, 366], "lines": [{"bbox": [93, 325, 541, 339], "spans": [{"bbox": [93, 325, 159, 339], "score": 1.0, "content": "Recall that ", "type": "text"}, {"bbox": [159, 327, 215, 337], "score": 0.93, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 10, "width": 56}, {"bbox": [216, 325, 401, 339], "score": 1.0, "content": ". Proposition 4.1(a) tells us that ", "type": "text"}, {"bbox": [401, 325, 484, 338], "score": 0.94, "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [484, 325, 541, 339], "score": 1.0, "content": ", for some", "type": "text"}], "index": 15}, {"bbox": [71, 339, 542, 354], "spans": [{"bbox": [71, 342, 88, 353], "score": 0.91, "content": "a,b", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [89, 339, 144, 354], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [144, 345, 151, 350], "score": 0.84, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [152, 339, 185, 354], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [185, 342, 200, 350], "score": 0.9, "content": "C^{a}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [201, 339, 464, 354], "score": 1.0, "content": ", we can assume without loss of generality that ", "type": "text"}, {"bbox": [464, 342, 497, 351], "score": 0.91, "content": "a\\;=\\;0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [497, 339, 542, 354], "score": 1.0, "content": ". Write", "type": "text"}], "index": 16}, {"bbox": [71, 353, 461, 369], "spans": [{"bbox": [71, 355, 137, 368], "score": 0.92, "content": "\\pi(J0)=J^{c}0", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [137, 353, 170, 369], "score": 1.0, "content": "; then ", "type": "text"}, {"bbox": [171, 359, 178, 365], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [178, 353, 312, 369], "score": 1.0, "content": " can be a permutation of ", "type": "text"}, {"bbox": [312, 356, 327, 367], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [328, 353, 369, 369], "score": 1.0, "content": " only if ", "type": "text"}, {"bbox": [369, 359, 375, 365], "score": 0.88, "content": "c", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [375, 353, 449, 369], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [450, 357, 456, 365], "score": 0.89, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [456, 353, 461, 369], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [88, 365, 518, 380], "lines": [{"bbox": [93, 366, 516, 384], "spans": [{"bbox": [93, 366, 106, 384], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 370, 136, 379], "score": 0.92, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [136, 366, 166, 384], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [167, 370, 281, 382], "score": 0.93, "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "type": "inline_equation", "height": 12, "width": 114}, {"bbox": [281, 366, 299, 384], "score": 1.0, "content": " so ", "type": "text"}, {"bbox": [299, 370, 363, 382], "score": 0.94, "content": "\\pi=\\pi[c-1]", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [363, 366, 482, 384], "score": 1.0, "content": ". Thus we can assume ", "type": "text"}, {"bbox": [482, 370, 511, 381], "score": 0.95, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [511, 366, 516, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [69, 381, 541, 530], "lines": [{"bbox": [93, 381, 541, 398], "spans": [{"bbox": [93, 381, 230, 398], "score": 1.0, "content": "Useful is the coefficient of ", "type": "text"}, {"bbox": [231, 384, 238, 393], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 381, 353, 398], "score": 1.0, "content": " in the tensor product ", "type": "text"}, {"bbox": [353, 384, 419, 395], "score": 0.91, "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 66}, {"bbox": [419, 381, 426, 398], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [426, 384, 468, 396], "score": 0.29, "content": "\\ell\\,\\mathrm{times})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [468, 381, 541, 398], "score": 1.0, "content": ": it is 0 unless", "type": "text"}], "index": 19}, {"bbox": [71, 393, 543, 419], "spans": [{"bbox": [71, 398, 115, 411], "score": 0.93, "content": "t(\\lambda)=\\ell", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [115, 393, 288, 419], "score": 1.0, "content": ", in which case the coefficient is", "type": "text"}, {"bbox": [288, 396, 309, 414], "score": 0.94, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 18, "width": 21}, {"bbox": [309, 393, 543, 419], "score": 1.0, "content": " (to get this, compare (3.1) above with [27,", "type": "text"}], "index": 20}, {"bbox": [70, 414, 541, 430], "spans": [{"bbox": [70, 414, 334, 430], "score": 1.0, "content": "p.114]) \u2014 we equate here the fundamental weights ", "type": "text"}, {"bbox": [335, 416, 349, 427], "score": 0.91, "content": "\\Lambda_{\\overline{{r}}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [349, 414, 373, 430], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [374, 417, 388, 428], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [388, 414, 430, 430], "score": 1.0, "content": ", so e.g. ", "type": "text"}, {"bbox": [430, 414, 458, 430], "score": 0.91, "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [458, 414, 498, 430], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [498, 416, 509, 426], "score": 0.39, "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [509, 414, 541, 430], "score": 1.0, "content": " when", "type": "text"}], "index": 21}, {"bbox": [71, 429, 541, 445], "spans": [{"bbox": [71, 432, 77, 440], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [78, 429, 120, 445], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [120, 431, 127, 440], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [128, 429, 167, 445], "score": 1.0, "content": ". Here, ", "type": "text"}, {"bbox": [167, 430, 242, 443], "score": 0.95, "content": "h(\\lambda)=\\prod h(x)", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [243, 429, 541, 445], "score": 1.0, "content": " is the product of the hook-lengths of the Young diagram", "type": "text"}], "index": 22}, {"bbox": [70, 443, 539, 460], "spans": [{"bbox": [70, 443, 162, 459], "score": 1.0, "content": "corresponding to ", "type": "text"}, {"bbox": [163, 446, 170, 454], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [171, 443, 381, 459], "score": 1.0, "content": ". Equation (2.4) tells us that as long as ", "type": "text"}, {"bbox": [381, 444, 446, 457], "score": 0.91, "content": "t(\\lambda)=\\ell\\leq k", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [447, 443, 517, 459], "score": 1.0, "content": ", the number", "type": "text"}, {"bbox": [518, 443, 539, 460], "score": 0.93, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 17, "width": 21}], "index": 23}, {"bbox": [69, 460, 542, 476], "spans": [{"bbox": [69, 460, 226, 476], "score": 1.0, "content": "will also be the coefficient of ", "type": "text"}, {"bbox": [227, 463, 243, 474], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [243, 460, 353, 476], "score": 1.0, "content": " in the fusion power ", "type": "text"}, {"bbox": [353, 461, 389, 475], "score": 0.92, "content": "(N_{\\Lambda_{1}})^{\\ell}", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [389, 460, 454, 476], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [454, 463, 506, 474], "score": 0.92, "content": "J0=k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [506, 460, 542, 476], "score": 1.0, "content": " is the", "type": "text"}], "index": 24}, {"bbox": [70, 475, 542, 491], "spans": [{"bbox": [70, 475, 348, 491], "score": 1.0, "content": "only simple-current appearing in the fusion product ", "type": "text"}, {"bbox": [348, 476, 362, 488], "score": 0.73, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [362, 475, 415, 491], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "type": "text"}, {"bbox": [415, 475, 430, 488], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [430, 475, 438, 491], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [439, 476, 446, 487], "score": 0.67, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [446, 475, 542, 491], "score": 1.0, "content": " times). Thus the", "type": "text"}], "index": 25}, {"bbox": [70, 489, 542, 505], "spans": [{"bbox": [70, 489, 354, 505], "score": 1.0, "content": "only nontrivial simple-current appearing in the fusion ", "type": "text"}, {"bbox": [354, 489, 376, 502], "score": 0.81, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [376, 489, 429, 505], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "type": "text"}, {"bbox": [429, 489, 451, 502], "score": 0.87, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [451, 489, 492, 505], "score": 1.0, "content": " will be", "type": "text"}, {"bbox": [493, 489, 525, 501], "score": 0.83, "content": "J^{b k}J0", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [526, 489, 542, 505], "score": 1.0, "content": " (0", "type": "text"}], "index": 26}, {"bbox": [72, 505, 541, 518], "spans": [{"bbox": [72, 505, 150, 518], "score": 1.0, "content": "will appear iff", "type": "text"}, {"bbox": [151, 507, 157, 515], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [158, 505, 203, 518], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [203, 506, 210, 515], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [210, 505, 262, 518], "score": 1.0, "content": "). Hence ", "type": "text"}, {"bbox": [263, 506, 322, 516], "score": 0.89, "content": "b k+1\\equiv c", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [323, 505, 358, 518], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [359, 506, 366, 515], "score": 0.59, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [366, 505, 484, 518], "score": 1.0, "content": ") must be coprime to ", "type": "text"}, {"bbox": [484, 506, 491, 515], "score": 0.76, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [492, 505, 541, 518], "score": 1.0, "content": ". This is", "type": "text"}], "index": 27}, {"bbox": [72, 519, 472, 533], "spans": [{"bbox": [72, 520, 251, 533], "score": 1.0, "content": "precisely the condition needed for ", "type": "text"}, {"bbox": [251, 519, 271, 532], "score": 0.92, "content": "\\pi[b]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [271, 520, 472, 533], "score": 1.0, "content": " to be a simple-current automorphism.", "type": "text"}], "index": 28}], "index": 23.5}, {"type": "text", "bbox": [70, 531, 541, 631], "lines": [{"bbox": [94, 532, 543, 547], "spans": [{"bbox": [94, 533, 289, 547], "score": 1.0, "content": "In other words, it suffices to consider ", "type": "text"}, {"bbox": [289, 533, 340, 545], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [340, 533, 398, 547], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [398, 532, 457, 546], "score": 0.92, "content": "\\pi[J0]=J0", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [457, 533, 543, 547], "score": 1.0, "content": ". We are done if", "type": "text"}], "index": 29}, {"bbox": [71, 547, 541, 563], "spans": [{"bbox": [71, 550, 99, 558], "score": 0.89, "content": "r=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [100, 547, 160, 563], "score": 1.0, "content": ", so assume ", "type": "text"}, {"bbox": [160, 549, 188, 560], "score": 0.89, "content": "r\\geq2", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [189, 547, 246, 563], "score": 1.0, "content": ". From the ", "type": "text"}, {"bbox": [246, 547, 261, 560], "score": 0.78, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [261, 547, 277, 563], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [278, 547, 293, 560], "score": 0.77, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [293, 547, 394, 563], "score": 1.0, "content": " fusion, we get that ", "type": "text"}, {"bbox": [394, 547, 481, 561], "score": 0.91, "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [482, 547, 541, 563], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 30}, {"bbox": [71, 561, 540, 576], "spans": [{"bbox": [71, 563, 91, 574], "score": 0.91, "content": "k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [92, 561, 431, 576], "score": 1.0, "content": " occurs (with multiplicity 1) in the tensor and fusion product of ", "type": "text"}, {"bbox": [431, 561, 452, 574], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [452, 561, 482, 576], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [482, 561, 534, 574], "score": 0.69, "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [534, 561, 540, 576], "score": 1.0, "content": "s,", "type": "text"}], "index": 31}, {"bbox": [70, 575, 541, 590], "spans": [{"bbox": [70, 576, 378, 590], "score": 1.0, "content": "but that it doesn\u2019t in the tensor (hence fusion) product of ", "type": "text"}, {"bbox": [379, 576, 393, 588], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [393, 576, 423, 590], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [423, 575, 469, 588], "score": 0.79, "content": "k-2~\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [470, 576, 541, 590], "score": 1.0, "content": "\u2019s (recall that", "type": "text"}], "index": 32}, {"bbox": [71, 589, 540, 606], "spans": [{"bbox": [71, 591, 186, 604], "score": 0.92, "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [187, 590, 425, 606], "score": 1.0, "content": " in the usual partial order on weights). Since ", "type": "text"}, {"bbox": [425, 590, 440, 603], "score": 0.72, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [440, 590, 456, 606], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [457, 590, 472, 603], "score": 0.74, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [472, 590, 524, 606], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "type": "text"}, {"bbox": [524, 589, 540, 603], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 16}], "index": 33}, {"bbox": [69, 603, 542, 621], "spans": [{"bbox": [69, 604, 163, 621], "score": 1.0, "content": "does not contain ", "type": "text"}, {"bbox": [164, 606, 178, 616], "score": 0.8, "content": "J0", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [178, 604, 185, 621], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [185, 603, 348, 618], "score": 0.25, "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 163}, {"bbox": [349, 604, 449, 621], "score": 1.0, "content": " should also avoid ", "type": "text"}, {"bbox": [449, 604, 513, 618], "score": 0.93, "content": "\\pi(J0)\\,=\\,J0", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [513, 604, 542, 621], "score": 1.0, "content": ", and", "type": "text"}], "index": 34}, {"bbox": [70, 618, 218, 634], "spans": [{"bbox": [70, 618, 97, 634], "score": 1.0, "content": "thus ", "type": "text"}, {"bbox": [97, 621, 118, 631], "score": 0.92, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [119, 618, 192, 634], "score": 1.0, "content": " cannot equal ", "type": "text"}, {"bbox": [193, 619, 213, 632], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 618, 218, 634], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 32}, {"type": "text", "bbox": [70, 632, 541, 703], "lines": [{"bbox": [94, 632, 543, 649], "spans": [{"bbox": [94, 632, 174, 649], "score": 1.0, "content": "Thus we know ", "type": "text"}, {"bbox": [174, 633, 226, 646], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [226, 632, 315, 649], "score": 1.0, "content": ". The remaining ", "type": "text"}, {"bbox": [316, 633, 366, 646], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [366, 632, 543, 649], "score": 1.0, "content": "follow quickly from induction: if", "type": "text"}], "index": 36}, {"bbox": [71, 647, 541, 663], "spans": [{"bbox": [71, 649, 121, 660], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [121, 647, 173, 663], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [173, 648, 224, 660], "score": 0.87, "content": "2\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [225, 647, 314, 663], "score": 1.0, "content": ", then the fusion ", "type": "text"}, {"bbox": [314, 647, 360, 660], "score": 0.44, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [360, 647, 404, 663], "score": 1.0, "content": "tells us ", "type": "text"}, {"bbox": [405, 648, 537, 662], "score": 0.93, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 14, "width": 132}, {"bbox": [537, 647, 541, 663], "score": 1.0, "content": ".", "type": "text"}], "index": 37}, {"bbox": [70, 661, 542, 678], "spans": [{"bbox": [70, 661, 95, 678], "score": 1.0, "content": "But ", "type": "text"}, {"bbox": [96, 662, 222, 675], "score": 0.92, "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [222, 661, 249, 678], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [250, 662, 347, 676], "score": 0.92, "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "type": "inline_equation", "height": 14, "width": 97}, {"bbox": [348, 661, 370, 678], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 663, 446, 675], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [446, 661, 487, 678], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [487, 664, 495, 673], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [495, 661, 542, 678], "score": 1.0, "content": " fixes all", "type": "text"}], "index": 38}, {"bbox": [70, 676, 542, 691], "spans": [{"bbox": [70, 676, 542, 691], "score": 1.0, "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "type": "text"}], "index": 39}, {"bbox": [70, 689, 384, 707], "spans": [{"bbox": [70, 689, 231, 707], "score": 1.0, "content": "the end of \u00a72.2) we know that ", "type": "text"}, {"bbox": [232, 695, 240, 701], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [240, 689, 362, 707], "score": 1.0, "content": " must fix everything in ", "type": "text"}, {"bbox": [362, 692, 378, 704], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [379, 689, 384, 707], "score": 1.0, "content": ".", "type": "text"}], "index": 40}], "index": 38}], "layout_bboxes": [], "page_idx": 16, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 314, 744], "spans": [{"bbox": [298, 731, 314, 744], "score": 1.0, "content": "17", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 542, 102], "lines": [{"bbox": [95, 74, 541, 88], "spans": [{"bbox": [95, 74, 541, 88], "score": 1.0, "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we", "type": "text"}], "index": 0}, {"bbox": [68, 85, 484, 106], "spans": [{"bbox": [68, 85, 264, 106], "score": 1.0, "content": "will sketch here the most interesting ", "type": "text"}, {"bbox": [265, 88, 296, 104], "score": 0.88, "content": "(A_{r}^{(1)})", "type": "inline_equation", "height": 16, "width": 31}, {"bbox": [297, 85, 416, 106], "score": 1.0, "content": " and the most difficult ", "type": "text"}, {"bbox": [417, 88, 449, 104], "score": 0.92, "content": "(E_{8}^{(1)})", "type": "inline_equation", "height": 16, "width": 32}, {"bbox": [449, 85, 484, 106], "score": 1.0, "content": " cases.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [68, 74, 541, 106]}, {"type": "text", "bbox": [70, 103, 541, 192], "lines": [{"bbox": [94, 104, 542, 121], "spans": [{"bbox": [94, 104, 167, 121], "score": 1.0, "content": "Consider first", "type": "text"}, {"bbox": [168, 106, 190, 119], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [190, 104, 263, 121], "score": 1.0, "content": ". By choosing ", "type": "text"}, {"bbox": [264, 106, 339, 119], "score": 0.91, "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [339, 104, 479, 121], "score": 1.0, "content": " in (4.1), we get that either ", "type": "text"}, {"bbox": [479, 106, 522, 117], "score": 0.93, "content": "\\lambda=k\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [522, 104, 542, 121], "score": 1.0, "content": "for", "type": "text"}], "index": 2}, {"bbox": [70, 119, 540, 135], "spans": [{"bbox": [70, 119, 100, 135], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [100, 121, 106, 129], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [106, 119, 182, 135], "score": 1.0, "content": ", in which case ", "type": "text"}, {"bbox": [182, 120, 190, 130], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 119, 335, 135], "score": 1.0, "content": " is a simple-current and (for ", "type": "text"}, {"bbox": [335, 120, 364, 132], "score": 0.9, "content": "k\\neq1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [364, 119, 371, 135], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [371, 120, 446, 133], "score": 0.93, "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [446, 119, 465, 135], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [465, 120, 540, 133], "score": 0.93, "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "type": "inline_equation", "height": 13, "width": 75}], "index": 3}, {"bbox": [70, 132, 541, 150], "spans": [{"bbox": [70, 132, 117, 150], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [117, 135, 123, 144], "score": 0.87, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [123, 132, 213, 150], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [214, 135, 256, 146], "score": 0.91, "content": "\\lambda\\in S\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [257, 132, 407, 150], "score": 1.0, "content": ". But then rank-level duality ", "type": "text"}, {"bbox": [407, 135, 494, 147], "score": 0.92, "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [495, 132, 541, 150], "score": 1.0, "content": " (defined", "type": "text"}], "index": 4}, {"bbox": [71, 148, 540, 165], "spans": [{"bbox": [71, 151, 104, 165], "score": 1.0, "content": "as for ", "type": "text"}, {"bbox": [104, 152, 126, 164], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [126, 151, 251, 165], "score": 1.0, "content": ", and which is exact for ", "type": "text"}, {"bbox": [252, 152, 273, 164], "score": 0.91, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [273, 151, 432, 165], "score": 1.0, "content": " q-dimensions) and (4.1) with ", "type": "text"}, {"bbox": [432, 148, 515, 162], "score": 0.94, "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [515, 151, 540, 165], "score": 1.0, "content": " give", "type": "text"}], "index": 5}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 87, 182], "score": 1.0, "content": "us ", "type": "text"}, {"bbox": [87, 165, 282, 180], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 195}, {"bbox": [282, 165, 380, 182], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [380, 168, 412, 177], "score": 0.92, "content": "\\ell\\,=\\,1", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [412, 165, 432, 182], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [433, 171, 439, 177], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [439, 165, 541, 182], "score": 1.0, "content": ". Com bining these", "type": "text"}], "index": 6}, {"bbox": [71, 181, 243, 196], "spans": [{"bbox": [71, 181, 243, 196], "score": 1.0, "content": "results yields Proposition 4.1(a).", "type": "text"}], "index": 7}], "index": 4.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [70, 104, 542, 196]}, {"type": "text", "bbox": [70, 193, 541, 288], "lines": [{"bbox": [93, 194, 541, 212], "spans": [{"bbox": [93, 194, 117, 212], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 197, 140, 209], "score": 0.93, "content": "E_{8,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 194, 244, 212], "score": 1.0, "content": ", run through each ", "type": "text"}, {"bbox": [244, 196, 357, 211], "score": 0.94, "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [358, 194, 541, 212], "score": 1.0, "content": " to reduce the proof to comparing", "type": "text"}], "index": 8}, {"bbox": [71, 209, 543, 232], "spans": [{"bbox": [71, 213, 104, 226], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [104, 209, 135, 232], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [136, 212, 181, 230], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [182, 209, 204, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [204, 214, 232, 225], "score": 0.91, "content": "i\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [233, 209, 255, 232], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [255, 213, 286, 226], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [287, 209, 309, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [309, 214, 349, 225], "score": 0.93, "content": "i\\neq0,1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [349, 209, 543, 232], "score": 1.0, "content": " (the argument in [18] unnecessarily", "type": "text"}], "index": 9}, {"bbox": [71, 228, 540, 244], "spans": [{"bbox": [71, 228, 540, 244], "score": 1.0, "content": "complicated things by restricting to integral weights). Standard arguments (see [18] for", "type": "text"}], "index": 10}, {"bbox": [69, 241, 543, 262], "spans": [{"bbox": [69, 241, 296, 262], "score": 1.0, "content": "details) quickly show that the q-dimension ", "type": "text"}, {"bbox": [297, 243, 342, 261], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [342, 241, 499, 262], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [499, 245, 506, 254], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [506, 241, 524, 262], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [524, 249, 536, 254], "score": 0.84, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 12}, {"bbox": [537, 241, 543, 262], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [72, 261, 542, 275], "spans": [{"bbox": [72, 261, 103, 275], "score": 1.0, "content": "while ", "type": "text"}, {"bbox": [104, 262, 135, 274], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [135, 261, 296, 275], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [297, 263, 304, 271], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [304, 261, 448, 275], "score": 1.0, "content": " to the Weyl dimension of ", "type": "text"}, {"bbox": [448, 263, 461, 273], "score": 0.92, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [461, 261, 542, 275], "score": 1.0, "content": ". The proof of", "type": "text"}], "index": 12}, {"bbox": [70, 276, 371, 290], "spans": [{"bbox": [70, 276, 371, 290], "score": 1.0, "content": "Proposition 4.1(e8) then reduces to a short computation.", "type": "text"}], "index": 13}], "index": 10.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [69, 194, 543, 290]}, {"type": "title", "bbox": [72, 301, 218, 316], "lines": [{"bbox": [71, 304, 219, 316], "spans": [{"bbox": [71, 304, 120, 316], "score": 1.0, "content": "4.2. The ", "type": "text"}, {"bbox": [121, 306, 130, 314], "score": 0.33, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [130, 304, 219, 316], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 323, 541, 366], "lines": [{"bbox": [93, 325, 541, 339], "spans": [{"bbox": [93, 325, 159, 339], "score": 1.0, "content": "Recall that ", "type": "text"}, {"bbox": [159, 327, 215, 337], "score": 0.93, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 10, "width": 56}, {"bbox": [216, 325, 401, 339], "score": 1.0, "content": ". Proposition 4.1(a) tells us that ", "type": "text"}, {"bbox": [401, 325, 484, 338], "score": 0.94, "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [484, 325, 541, 339], "score": 1.0, "content": ", for some", "type": "text"}], "index": 15}, {"bbox": [71, 339, 542, 354], "spans": [{"bbox": [71, 342, 88, 353], "score": 0.91, "content": "a,b", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [89, 339, 144, 354], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [144, 345, 151, 350], "score": 0.84, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [152, 339, 185, 354], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [185, 342, 200, 350], "score": 0.9, "content": "C^{a}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [201, 339, 464, 354], "score": 1.0, "content": ", we can assume without loss of generality that ", "type": "text"}, {"bbox": [464, 342, 497, 351], "score": 0.91, "content": "a\\;=\\;0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [497, 339, 542, 354], "score": 1.0, "content": ". Write", "type": "text"}], "index": 16}, {"bbox": [71, 353, 461, 369], "spans": [{"bbox": [71, 355, 137, 368], "score": 0.92, "content": "\\pi(J0)=J^{c}0", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [137, 353, 170, 369], "score": 1.0, "content": "; then ", "type": "text"}, {"bbox": [171, 359, 178, 365], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [178, 353, 312, 369], "score": 1.0, "content": " can be a permutation of ", "type": "text"}, {"bbox": [312, 356, 327, 367], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [328, 353, 369, 369], "score": 1.0, "content": " only if ", "type": "text"}, {"bbox": [369, 359, 375, 365], "score": 0.88, "content": "c", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [375, 353, 449, 369], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [450, 357, 456, 365], "score": 0.89, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [456, 353, 461, 369], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [71, 325, 542, 369]}, {"type": "text", "bbox": [88, 365, 518, 380], "lines": [{"bbox": [93, 366, 516, 384], "spans": [{"bbox": [93, 366, 106, 384], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 370, 136, 379], "score": 0.92, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [136, 366, 166, 384], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [167, 370, 281, 382], "score": 0.93, "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "type": "inline_equation", "height": 12, "width": 114}, {"bbox": [281, 366, 299, 384], "score": 1.0, "content": " so ", "type": "text"}, {"bbox": [299, 370, 363, 382], "score": 0.94, "content": "\\pi=\\pi[c-1]", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [363, 366, 482, 384], "score": 1.0, "content": ". Thus we can assume ", "type": "text"}, {"bbox": [482, 370, 511, 381], "score": 0.95, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [511, 366, 516, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [93, 366, 516, 384]}, {"type": "text", "bbox": [69, 381, 541, 530], "lines": [{"bbox": [93, 381, 541, 398], "spans": [{"bbox": [93, 381, 230, 398], "score": 1.0, "content": "Useful is the coefficient of ", "type": "text"}, {"bbox": [231, 384, 238, 393], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 381, 353, 398], "score": 1.0, "content": " in the tensor product ", "type": "text"}, {"bbox": [353, 384, 419, 395], "score": 0.91, "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 66}, {"bbox": [419, 381, 426, 398], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [426, 384, 468, 396], "score": 0.29, "content": "\\ell\\,\\mathrm{times})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [468, 381, 541, 398], "score": 1.0, "content": ": it is 0 unless", "type": "text"}], "index": 19}, {"bbox": [71, 393, 543, 419], "spans": [{"bbox": [71, 398, 115, 411], "score": 0.93, "content": "t(\\lambda)=\\ell", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [115, 393, 288, 419], "score": 1.0, "content": ", in which case the coefficient is", "type": "text"}, {"bbox": [288, 396, 309, 414], "score": 0.94, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 18, "width": 21}, {"bbox": [309, 393, 543, 419], "score": 1.0, "content": " (to get this, compare (3.1) above with [27,", "type": "text"}], "index": 20}, {"bbox": [70, 414, 541, 430], "spans": [{"bbox": [70, 414, 334, 430], "score": 1.0, "content": "p.114]) \u2014 we equate here the fundamental weights ", "type": "text"}, {"bbox": [335, 416, 349, 427], "score": 0.91, "content": "\\Lambda_{\\overline{{r}}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [349, 414, 373, 430], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [374, 417, 388, 428], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [388, 414, 430, 430], "score": 1.0, "content": ", so e.g. ", "type": "text"}, {"bbox": [430, 414, 458, 430], "score": 0.91, "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [458, 414, 498, 430], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [498, 416, 509, 426], "score": 0.39, "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [509, 414, 541, 430], "score": 1.0, "content": " when", "type": "text"}], "index": 21}, {"bbox": [71, 429, 541, 445], "spans": [{"bbox": [71, 432, 77, 440], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [78, 429, 120, 445], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [120, 431, 127, 440], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [128, 429, 167, 445], "score": 1.0, "content": ". Here, ", "type": "text"}, {"bbox": [167, 430, 242, 443], "score": 0.95, "content": "h(\\lambda)=\\prod h(x)", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [243, 429, 541, 445], "score": 1.0, "content": " is the product of the hook-lengths of the Young diagram", "type": "text"}], "index": 22}, {"bbox": [70, 443, 539, 460], "spans": [{"bbox": [70, 443, 162, 459], "score": 1.0, "content": "corresponding to ", "type": "text"}, {"bbox": [163, 446, 170, 454], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [171, 443, 381, 459], "score": 1.0, "content": ". Equation (2.4) tells us that as long as ", "type": "text"}, {"bbox": [381, 444, 446, 457], "score": 0.91, "content": "t(\\lambda)=\\ell\\leq k", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [447, 443, 517, 459], "score": 1.0, "content": ", the number", "type": "text"}, {"bbox": [518, 443, 539, 460], "score": 0.93, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 17, "width": 21}], "index": 23}, {"bbox": [69, 460, 542, 476], "spans": [{"bbox": [69, 460, 226, 476], "score": 1.0, "content": "will also be the coefficient of ", "type": "text"}, {"bbox": [227, 463, 243, 474], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [243, 460, 353, 476], "score": 1.0, "content": " in the fusion power ", "type": "text"}, {"bbox": [353, 461, 389, 475], "score": 0.92, "content": "(N_{\\Lambda_{1}})^{\\ell}", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [389, 460, 454, 476], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [454, 463, 506, 474], "score": 0.92, "content": "J0=k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [506, 460, 542, 476], "score": 1.0, "content": " is the", "type": "text"}], "index": 24}, {"bbox": [70, 475, 542, 491], "spans": [{"bbox": [70, 475, 348, 491], "score": 1.0, "content": "only simple-current appearing in the fusion product ", "type": "text"}, {"bbox": [348, 476, 362, 488], "score": 0.73, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [362, 475, 415, 491], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "type": "text"}, {"bbox": [415, 475, 430, 488], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [430, 475, 438, 491], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [439, 476, 446, 487], "score": 0.67, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [446, 475, 542, 491], "score": 1.0, "content": " times). Thus the", "type": "text"}], "index": 25}, {"bbox": [70, 489, 542, 505], "spans": [{"bbox": [70, 489, 354, 505], "score": 1.0, "content": "only nontrivial simple-current appearing in the fusion ", "type": "text"}, {"bbox": [354, 489, 376, 502], "score": 0.81, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [376, 489, 429, 505], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "type": "text"}, {"bbox": [429, 489, 451, 502], "score": 0.87, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [451, 489, 492, 505], "score": 1.0, "content": " will be", "type": "text"}, {"bbox": [493, 489, 525, 501], "score": 0.83, "content": "J^{b k}J0", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [526, 489, 542, 505], "score": 1.0, "content": " (0", "type": "text"}], "index": 26}, {"bbox": [72, 505, 541, 518], "spans": [{"bbox": [72, 505, 150, 518], "score": 1.0, "content": "will appear iff", "type": "text"}, {"bbox": [151, 507, 157, 515], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [158, 505, 203, 518], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [203, 506, 210, 515], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [210, 505, 262, 518], "score": 1.0, "content": "). Hence ", "type": "text"}, {"bbox": [263, 506, 322, 516], "score": 0.89, "content": "b k+1\\equiv c", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [323, 505, 358, 518], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [359, 506, 366, 515], "score": 0.59, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [366, 505, 484, 518], "score": 1.0, "content": ") must be coprime to ", "type": "text"}, {"bbox": [484, 506, 491, 515], "score": 0.76, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [492, 505, 541, 518], "score": 1.0, "content": ". This is", "type": "text"}], "index": 27}, {"bbox": [72, 519, 472, 533], "spans": [{"bbox": [72, 520, 251, 533], "score": 1.0, "content": "precisely the condition needed for ", "type": "text"}, {"bbox": [251, 519, 271, 532], "score": 0.92, "content": "\\pi[b]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [271, 520, 472, 533], "score": 1.0, "content": " to be a simple-current automorphism.", "type": "text"}], "index": 28}], "index": 23.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [69, 381, 543, 533]}, {"type": "text", "bbox": [70, 531, 541, 631], "lines": [{"bbox": [94, 532, 543, 547], "spans": [{"bbox": [94, 533, 289, 547], "score": 1.0, "content": "In other words, it suffices to consider ", "type": "text"}, {"bbox": [289, 533, 340, 545], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [340, 533, 398, 547], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [398, 532, 457, 546], "score": 0.92, "content": "\\pi[J0]=J0", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [457, 533, 543, 547], "score": 1.0, "content": ". We are done if", "type": "text"}], "index": 29}, {"bbox": [71, 547, 541, 563], "spans": [{"bbox": [71, 550, 99, 558], "score": 0.89, "content": "r=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [100, 547, 160, 563], "score": 1.0, "content": ", so assume ", "type": "text"}, {"bbox": [160, 549, 188, 560], "score": 0.89, "content": "r\\geq2", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [189, 547, 246, 563], "score": 1.0, "content": ". From the ", "type": "text"}, {"bbox": [246, 547, 261, 560], "score": 0.78, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [261, 547, 277, 563], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [278, 547, 293, 560], "score": 0.77, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [293, 547, 394, 563], "score": 1.0, "content": " fusion, we get that ", "type": "text"}, {"bbox": [394, 547, 481, 561], "score": 0.91, "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [482, 547, 541, 563], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 30}, {"bbox": [71, 561, 540, 576], "spans": [{"bbox": [71, 563, 91, 574], "score": 0.91, "content": "k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [92, 561, 431, 576], "score": 1.0, "content": " occurs (with multiplicity 1) in the tensor and fusion product of ", "type": "text"}, {"bbox": [431, 561, 452, 574], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [452, 561, 482, 576], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [482, 561, 534, 574], "score": 0.69, "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [534, 561, 540, 576], "score": 1.0, "content": "s,", "type": "text"}], "index": 31}, {"bbox": [70, 575, 541, 590], "spans": [{"bbox": [70, 576, 378, 590], "score": 1.0, "content": "but that it doesn\u2019t in the tensor (hence fusion) product of ", "type": "text"}, {"bbox": [379, 576, 393, 588], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [393, 576, 423, 590], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [423, 575, 469, 588], "score": 0.79, "content": "k-2~\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [470, 576, 541, 590], "score": 1.0, "content": "\u2019s (recall that", "type": "text"}], "index": 32}, {"bbox": [71, 589, 540, 606], "spans": [{"bbox": [71, 591, 186, 604], "score": 0.92, "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [187, 590, 425, 606], "score": 1.0, "content": " in the usual partial order on weights). Since ", "type": "text"}, {"bbox": [425, 590, 440, 603], "score": 0.72, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [440, 590, 456, 606], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [457, 590, 472, 603], "score": 0.74, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [472, 590, 524, 606], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "type": "text"}, {"bbox": [524, 589, 540, 603], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 16}], "index": 33}, {"bbox": [69, 603, 542, 621], "spans": [{"bbox": [69, 604, 163, 621], "score": 1.0, "content": "does not contain ", "type": "text"}, {"bbox": [164, 606, 178, 616], "score": 0.8, "content": "J0", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [178, 604, 185, 621], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [185, 603, 348, 618], "score": 0.25, "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 163}, {"bbox": [349, 604, 449, 621], "score": 1.0, "content": " should also avoid ", "type": "text"}, {"bbox": [449, 604, 513, 618], "score": 0.93, "content": "\\pi(J0)\\,=\\,J0", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [513, 604, 542, 621], "score": 1.0, "content": ", and", "type": "text"}], "index": 34}, {"bbox": [70, 618, 218, 634], "spans": [{"bbox": [70, 618, 97, 634], "score": 1.0, "content": "thus ", "type": "text"}, {"bbox": [97, 621, 118, 631], "score": 0.92, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [119, 618, 192, 634], "score": 1.0, "content": " cannot equal ", "type": "text"}, {"bbox": [193, 619, 213, 632], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 618, 218, 634], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 32, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [69, 532, 543, 634]}, {"type": "text", "bbox": [70, 632, 541, 703], "lines": [{"bbox": [94, 632, 543, 649], "spans": [{"bbox": [94, 632, 174, 649], "score": 1.0, "content": "Thus we know ", "type": "text"}, {"bbox": [174, 633, 226, 646], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [226, 632, 315, 649], "score": 1.0, "content": ". The remaining ", "type": "text"}, {"bbox": [316, 633, 366, 646], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [366, 632, 543, 649], "score": 1.0, "content": "follow quickly from induction: if", "type": "text"}], "index": 36}, {"bbox": [71, 647, 541, 663], "spans": [{"bbox": [71, 649, 121, 660], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [121, 647, 173, 663], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [173, 648, 224, 660], "score": 0.87, "content": "2\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [225, 647, 314, 663], "score": 1.0, "content": ", then the fusion ", "type": "text"}, {"bbox": [314, 647, 360, 660], "score": 0.44, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [360, 647, 404, 663], "score": 1.0, "content": "tells us ", "type": "text"}, {"bbox": [405, 648, 537, 662], "score": 0.93, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 14, "width": 132}, {"bbox": [537, 647, 541, 663], "score": 1.0, "content": ".", "type": "text"}], "index": 37}, {"bbox": [70, 661, 542, 678], "spans": [{"bbox": [70, 661, 95, 678], "score": 1.0, "content": "But ", "type": "text"}, {"bbox": [96, 662, 222, 675], "score": 0.92, "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [222, 661, 249, 678], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [250, 662, 347, 676], "score": 0.92, "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "type": "inline_equation", "height": 14, "width": 97}, {"bbox": [348, 661, 370, 678], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 663, 446, 675], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [446, 661, 487, 678], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [487, 664, 495, 673], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [495, 661, 542, 678], "score": 1.0, "content": " fixes all", "type": "text"}], "index": 38}, {"bbox": [70, 676, 542, 691], "spans": [{"bbox": [70, 676, 542, 691], "score": 1.0, "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "type": "text"}], "index": 39}, {"bbox": [70, 689, 384, 707], "spans": [{"bbox": [70, 689, 231, 707], "score": 1.0, "content": "the end of \u00a72.2) we know that ", "type": "text"}, {"bbox": [232, 695, 240, 701], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [240, 689, 362, 707], "score": 1.0, "content": " must fix everything in ", "type": "text"}, {"bbox": [362, 692, 378, 704], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [379, 689, 384, 707], "score": 1.0, "content": ".", "type": "text"}], "index": 40}], "index": 38, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [70, 632, 543, 707]}]}
0002044v1	20	# 5. Affine fusion ring isomorphisms We conclude the paper with the determination of all isomorphisms among the affine fusion rings $$\mathcal{R}(X_{r,k})$$ . Recall Definition 2.1 and the discussion in $$\S2.2$$ . Theorem 5.1. The complete list of fusion ring isomorphisms $$\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(Y_{s,m})$$ when $$X_{r,k}\neq Y_{s,m}$$ (where $$X_{r},Y_{s}$$ are simple) is: rank-level duality $$\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$$ for all $$r,k$$ , as well as $$\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$$ ; $$\mathcal{R}(B_{r,1})\cong\mathcal{R}(A_{1,2})\cong\mathcal{R}(C_{2,1})\cong\mathcal{R}(E_{8,2})$$ for all $$r\geq3$$ ; $$\mathcal{R}(A_{3,1})\cong\mathcal{R}(D_{o d d,1})$$ ; $$\mathcal{R}(D_{r,1})\cong\mathcal{R}(D_{s,1})$$ whenever $$r\equiv s$$ (mod 2); $$\mathscr{R}(A_{2,1})\cong\mathscr{R}(E_{6,1})$$ and $$\mathcal{R}(A_{1,1})\cong\mathcal{R}(E_{7,1})$$ ; $$\mathcal{R}(F_{4,1})\cong\mathcal{R}(G_{2,1})$$ , $$\mathcal{R}(F_{4,2})\cong\mathcal{R}(E_{8,3})$$ , and $$\mathcal{R}(F_{4,3})\cong\mathcal{R}(G_{2,4})$$ . The isomorphism $$\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$$ takes $$a\Lambda_{1}$$ to $$\widetilde{\Lambda}_{a}$$ . The isomorphism $$\mathcal{R}(F_{4,2})\cong$$ $$\mathcal{R}(E_{8,3})$$ was found in [14]; it relates $$\Lambda_{1}\leftrightarrow\tilde{\Lambda}_{8}$$ , $$2\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{2}$$ , $$\Lambda_{3}\,\leftrightarrow\,\widetilde{\Lambda}_{1}$$ , $$\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{7}$$ . The isomorphism $$\mathcal{R}(F_{4,3})\,\cong\,\mathcal{R}(G_{2,4})$$ was found i n [34,14]; a corresponde nce which works is $$\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}$$ , $$\Lambda_{1}\leftrightarrow2\widetilde{\Lambda}_{1}$$ , $$\Lambda_{3}\leftrightarrow3\widetilde{\Lambda}_{2}$$ , $$2\Lambda_{4}\leftrightarrow2\widetilde{\Lambda}_{2}$$ , $$\Lambda_{1}+\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}+2\widetilde{\Lambda}_{2}$$ , $$\Lambda_{2}\leftrightarrow4\widetilde{\Lambda}_{2}$$ , $$3\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{2}$$ , and $$\Lambda_{3}+\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{1}+\tilde{\Lambda}_{2}$$ . We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $$\mathcal{R}(A_{r,k})$$ and $$\mathcal{R}(A_{s,m})$$ are isomorphic. Then their simple-current groups $$\mathbb{Z}_{r+1}$$ and $$\mathbb{Z}_{s+1}$$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $$r=s$$ . Now compare the numbers $$\|\|P_{+}\|\|$$ of highest-weights: $$\big(\begin{array}{c}{{r+k}}\\ {{r}}\end{array}\big)=\big(\begin{array}{c}{{r+m}}\\ {{r}}\end{array}\big)$$ , which forces $$m=k$$ . It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $$P_{+}^{k}(X_{r}^{(1)})$$ with second smallest q-dimension are precisely the orbit $$S\Lambda_{\star}$$ , except for: $$A_{r,1}$$ ; $$B_{r,k}$$ for $$k\leq3$$ ; $$C_{2,2},C_{2,3},C_{3,2}$$ ; $$D_{r,k}$$ for $$k\leq2$$ ; $$E_{6,k}$$ for $$k\leq2$$ ; and $$E_{7,k},E_{8,k},F_{4,k},G_{2,k}$$ for $$k\leq4$$ . $$C_{r,k}$$ and $$B_{s,m}$$ both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit $$J^{i}\Lambda_{1}$$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $$Q_{j}(J^{i}\Lambda_{1})$$ for the two algebras are different. Another useful invariant involves the set of integers $$\ell$$ coprime to $$\kappa N$$ for which $$0^{(\ell)}$$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $$0^{(\ell)}$$ ( $$\ell$$ coprime to $$2\kappa$$ ) for the algebras $$B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$$ is, respectively,	<h1>5. Affine fusion ring isomorphisms</h1> <p>We conclude the paper with the determination of all isomorphisms among the affine fusion rings $$\mathcal{R}(X_{r,k})$$ . Recall Definition 2.1 and the discussion in $$\S2.2$$ .</p> <p>Theorem 5.1. The complete list of fusion ring isomorphisms $$\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(Y_{s,m})$$ when $$X_{r,k}\neq Y_{s,m}$$ (where $$X_{r},Y_{s}$$ are simple) is: rank-level duality $$\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$$ for all $$r,k$$ , as well as $$\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$$ ; $$\mathcal{R}(B_{r,1})\cong\mathcal{R}(A_{1,2})\cong\mathcal{R}(C_{2,1})\cong\mathcal{R}(E_{8,2})$$ for all $$r\geq3$$ ; $$\mathcal{R}(A_{3,1})\cong\mathcal{R}(D_{o d d,1})$$ ; $$\mathcal{R}(D_{r,1})\cong\mathcal{R}(D_{s,1})$$ whenever $$r\equiv s$$ (mod 2); $$\mathscr{R}(A_{2,1})\cong\mathscr{R}(E_{6,1})$$ and $$\mathcal{R}(A_{1,1})\cong\mathcal{R}(E_{7,1})$$ ; $$\mathcal{R}(F_{4,1})\cong\mathcal{R}(G_{2,1})$$ , $$\mathcal{R}(F_{4,2})\cong\mathcal{R}(E_{8,3})$$ , and $$\mathcal{R}(F_{4,3})\cong\mathcal{R}(G_{2,4})$$ .</p> <p>The isomorphism $$\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$$ takes $$a\Lambda_{1}$$ to $$\widetilde{\Lambda}_{a}$$ . The isomorphism $$\mathcal{R}(F_{4,2})\cong$$ $$\mathcal{R}(E_{8,3})$$ was found in [14]; it relates $$\Lambda_{1}\leftrightarrow\tilde{\Lambda}_{8}$$ , $$2\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{2}$$ , $$\Lambda_{3}\,\leftrightarrow\,\widetilde{\Lambda}_{1}$$ , $$\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{7}$$ . The isomorphism $$\mathcal{R}(F_{4,3})\,\cong\,\mathcal{R}(G_{2,4})$$ was found i n [34,14]; a corresponde nce which works is $$\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}$$ , $$\Lambda_{1}\leftrightarrow2\widetilde{\Lambda}_{1}$$ , $$\Lambda_{3}\leftrightarrow3\widetilde{\Lambda}_{2}$$ , $$2\Lambda_{4}\leftrightarrow2\widetilde{\Lambda}_{2}$$ , $$\Lambda_{1}+\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}+2\widetilde{\Lambda}_{2}$$ , $$\Lambda_{2}\leftrightarrow4\widetilde{\Lambda}_{2}$$ , $$3\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{2}$$ , and $$\Lambda_{3}+\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{1}+\tilde{\Lambda}_{2}$$ .</p> <p>We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $$\mathcal{R}(A_{r,k})$$ and $$\mathcal{R}(A_{s,m})$$ are isomorphic. Then their simple-current groups $$\mathbb{Z}_{r+1}$$ and $$\mathbb{Z}_{s+1}$$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $$r=s$$ . Now compare the numbers $$\|\|P_{+}\|\|$$ of highest-weights: $$\big(\begin{array}{c}{{r+k}}\\ {{r}}\end{array}\big)=\big(\begin{array}{c}{{r+m}}\\ {{r}}\end{array}\big)$$ , which forces $$m=k$$ .</p> <p>It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $$P_{+}^{k}(X_{r}^{(1)})$$ with second smallest q-dimension are precisely the orbit $$S\Lambda_{\star}$$ , except for: $$A_{r,1}$$ ; $$B_{r,k}$$ for $$k\leq3$$ ; $$C_{2,2},C_{2,3},C_{3,2}$$ ; $$D_{r,k}$$ for $$k\leq2$$ ; $$E_{6,k}$$ for $$k\leq2$$ ; and $$E_{7,k},E_{8,k},F_{4,k},G_{2,k}$$ for $$k\leq4$$ .</p> <p>$$C_{r,k}$$ and $$B_{s,m}$$ both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit $$J^{i}\Lambda_{1}$$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $$Q_{j}(J^{i}\Lambda_{1})$$ for the two algebras are different.</p> <p>Another useful invariant involves the set of integers $$\ell$$ coprime to $$\kappa N$$ for which $$0^{(\ell)}$$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $$0^{(\ell)}$$ ( $$\ell$$ coprime to $$2\kappa$$ ) for the algebras $$B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$$ is, respectively,</p>	[{"type": "title", "coordinates": [200, 71, 410, 86], "content": "5. Affine fusion ring isomorphisms", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [70, 102, 542, 132], "content": "We conclude the paper with the determination of all isomorphisms among the affine\nfusion rings $$\\mathcal{R}(X_{r,k})$$ . Recall Definition 2.1 and the discussion in $$\\S2.2$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 138, 541, 259], "content": "Theorem 5.1. The complete list of fusion ring isomorphisms $$\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})$$\nwhen $$X_{r,k}\\neq Y_{s,m}$$ (where $$X_{r},Y_{s}$$ are simple) is:\nrank-level duality $$\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$$ for all $$r,k$$ , as well as $$\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$$ ;\n$$\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})$$ for all $$r\\geq3$$ ;\n$$\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})$$ ;\n$$\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})$$ whenever $$r\\equiv s$$ (mod 2);\n$$\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})$$ and $$\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})$$ ;\n$$\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})$$ , $$\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})$$ , and $$\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 266, 541, 343], "content": "The isomorphism $$\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$$ takes $$a\\Lambda_{1}$$ to $$\\widetilde{\\Lambda}_{a}$$ . The isomorphism $$\\mathcal{R}(F_{4,2})\\cong$$\n$$\\mathcal{R}(E_{8,3})$$ was found in [14]; it relates $$\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}$$ , $$2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}$$ , $$\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}$$ , $$\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}$$ . The\nisomorphism $$\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})$$ was found i n [34,14]; a corresponde nce which works is\n$$\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}$$ , $$\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}$$ , $$\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}$$ , $$2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}$$ , $$\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}$$ , $$\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}$$ , $$3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}$$ ,\nand $$\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 344, 541, 416], "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion\nrings, case by case. For example, suppose $$\\mathcal{R}(A_{r,k})$$ and $$\\mathcal{R}(A_{s,m})$$ are isomorphic. Then their\nsimple-current groups $$\\mathbb{Z}_{r+1}$$ and $$\\mathbb{Z}_{s+1}$$ must be isomorphic (since simple-currents must get\nmapped to simple-currents), so $$r=s$$ . Now compare the numbers $$\|\|P_{+}\|\|$$ of highest-weights:\n$$\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)$$ , which forces $$m=k$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [70, 416, 541, 490], "content": "It is also quite useful here to know those weights with second smallest q-dimension.\nThis is a by-product of the proof of Proposition 4.1, and the complete answer is given in\n[18, Table 3]. Here we will simply state that those weights in $$P_{+}^{k}(X_{r}^{(1)})$$ with second smallest\nq-dimension are precisely the orbit $$S\\Lambda_{\\star}$$ , except for: $$A_{r,1}$$ ; $$B_{r,k}$$ for $$k\\leq3$$ ; $$C_{2,2},C_{2,3},C_{3,2}$$ ;\n$$D_{r,k}$$ for $$k\\leq2$$ ; $$E_{6,k}$$ for $$k\\leq2$$ ; and $$E_{7,k},E_{8,k},F_{4,k},G_{2,k}$$ for $$k\\leq4$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [71, 490, 541, 533], "content": "$$C_{r,k}$$ and $$B_{s,m}$$ both have two simple-currents, but their fusion rings can\u2019t be isomorphic\n(generically) because the orbit $$J^{i}\\Lambda_{1}$$ has the second smallest q-dimension for both algebras\nat generic rank/level, but the numbers $$Q_{j}(J^{i}\\Lambda_{1})$$ for the two algebras are different.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 534, 541, 579], "content": "Another useful invariant involves the set of integers $$\\ell$$ coprime to $$\\kappa N$$ for which $$0^{(\\ell)}$$ is\na simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,\nthe q-dimension of $$0^{(\\ell)}$$ ( $$\\ell$$ coprime to $$2\\kappa$$ ) for the algebras $$B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$$ is, respectively,", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [145, 595, 464, 720], "content": "", "block_type": "interline_equation", "index": 9}]	[{"type": "text", "coordinates": [201, 74, 409, 88], "content": "5. Affine fusion ring isomorphisms", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [94, 103, 541, 120], "content": "We conclude the paper with the determination of all isomorphisms among the affine", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [72, 119, 135, 133], "content": "fusion rings ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [135, 120, 178, 133], "content": "\\mathcal{R}(X_{r,k})", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [178, 119, 410, 133], "content": ". Recall Definition 2.1 and the discussion in ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [411, 119, 432, 132], "content": "\\S2.2", "score": 0.3, "index": 6}, {"type": "text", "coordinates": [433, 119, 435, 133], "content": ".", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [94, 140, 434, 159], "content": "Theorem 5.1. The complete list of fusion ring isomorphisms ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [434, 143, 540, 156], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [72, 156, 101, 172], "content": "when ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [101, 156, 164, 170], "content": "X_{r,k}\\neq Y_{s,m}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [164, 156, 206, 172], "content": " (where ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [206, 156, 240, 169], "content": "X_{r},Y_{s}", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [240, 156, 321, 172], "content": " are simple) is:", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [70, 169, 164, 187], "content": "rank-level duality ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [164, 171, 263, 185], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "score": 0.91, "index": 16}, {"type": "text", "coordinates": [263, 169, 302, 187], "content": " for all ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [303, 171, 321, 184], "content": "r,k", "score": 0.88, "index": 18}, {"type": "text", "coordinates": [321, 169, 382, 187], "content": ", as well as ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [383, 171, 483, 185], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "score": 0.92, "index": 20}, {"type": "text", "coordinates": [484, 169, 489, 187], "content": ";", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [71, 185, 284, 200], "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "score": 0.86, "index": 22}, {"type": "text", "coordinates": [284, 185, 323, 203], "content": " for all ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [323, 186, 352, 199], "content": "r\\geq3", "score": 0.89, "index": 24}, {"type": "text", "coordinates": [353, 185, 358, 203], "content": ";", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [71, 200, 182, 214], "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "score": 0.89, "index": 26}, {"type": "text", "coordinates": [182, 200, 188, 217], "content": ";", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [71, 215, 171, 229], "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "score": 0.9, "index": 28}, {"type": "text", "coordinates": [171, 214, 226, 231], "content": " whenever ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [226, 216, 255, 227], "content": "r\\equiv s", "score": 0.48, "index": 30}, {"type": "text", "coordinates": [256, 214, 307, 231], "content": " (mod 2);", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [71, 230, 171, 244], "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "score": 0.89, "index": 32}, {"type": "text", "coordinates": [171, 230, 197, 245], "content": " and ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [197, 229, 296, 244], "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "score": 0.87, "index": 34}, {"type": "text", "coordinates": [296, 230, 302, 245], "content": ";", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [71, 245, 169, 259], "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "score": 0.89, "index": 36}, {"type": "text", "coordinates": [170, 244, 176, 261], "content": ", ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [177, 244, 275, 259], "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "score": 0.85, "index": 38}, {"type": "text", "coordinates": [275, 244, 304, 261], "content": ", and", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [305, 244, 405, 259], "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "score": 0.9, "index": 40}, {"type": "text", "coordinates": [405, 244, 408, 261], "content": ".", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [93, 267, 190, 285], "content": "The isomorphism ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [190, 269, 291, 284], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [292, 267, 326, 285], "content": " takes ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [326, 269, 347, 282], "content": "a\\Lambda_{1}", "score": 0.86, "index": 45}, {"type": "text", "coordinates": [348, 267, 365, 285], "content": " to", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [366, 267, 381, 282], "content": "\\widetilde{\\Lambda}_{a}", "score": 0.9, "index": 47}, {"type": "text", "coordinates": [381, 267, 485, 285], "content": ". The isomorphism ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [486, 270, 541, 284], "content": "\\mathcal{R}(F_{4,2})\\cong", "score": 0.89, "index": 49}, {"type": "inline_equation", "coordinates": [71, 288, 113, 300], "content": "\\mathcal{R}(E_{8,3})", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [113, 285, 272, 300], "content": " was found in [14]; it relates ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [272, 283, 323, 299], "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [323, 285, 331, 300], "content": ", ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [331, 284, 388, 299], "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "score": 0.88, "index": 54}, {"type": "text", "coordinates": [389, 285, 396, 300], "content": ", ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [397, 285, 447, 299], "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "score": 0.93, "index": 56}, {"type": "text", "coordinates": [448, 285, 455, 300], "content": ", ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [456, 285, 506, 299], "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "score": 0.93, "index": 58}, {"type": "text", "coordinates": [507, 285, 540, 300], "content": ". The", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [70, 299, 142, 315], "content": "isomorphism ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [142, 300, 244, 314], "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "score": 0.91, "index": 61}, {"type": "text", "coordinates": [244, 299, 542, 315], "content": " was found i n [34,14]; a corresponde nce which works is", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [71, 315, 118, 330], "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [118, 315, 124, 331], "content": ", ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [124, 315, 177, 329], "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "score": 0.88, "index": 65}, {"type": "text", "coordinates": [177, 315, 183, 331], "content": ", ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [184, 315, 236, 330], "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "score": 0.87, "index": 67}, {"type": "text", "coordinates": [236, 315, 242, 331], "content": ",", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [243, 314, 302, 329], "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "score": 0.9, "index": 69}, {"type": "text", "coordinates": [302, 315, 308, 331], "content": ", ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [308, 315, 417, 330], "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "score": 0.89, "index": 71}, {"type": "text", "coordinates": [418, 315, 424, 331], "content": ", ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [424, 315, 477, 330], "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "score": 0.9, "index": 73}, {"type": "text", "coordinates": [477, 315, 483, 331], "content": ", ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [484, 315, 536, 329], "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [536, 315, 540, 331], "content": ",", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [70, 329, 94, 345], "content": "and ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [95, 330, 197, 344], "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "score": 0.91, "index": 78}, {"type": "text", "coordinates": [198, 329, 201, 345], "content": ".", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [95, 345, 540, 361], "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [69, 359, 286, 376], "content": "rings, case by case. For example, suppose ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [286, 361, 328, 374], "content": "\\mathcal{R}(A_{r,k})", "score": 0.94, "index": 82}, {"type": "text", "coordinates": [328, 359, 353, 376], "content": " and ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [353, 361, 398, 374], "content": "\\mathcal{R}(A_{s,m})", "score": 0.94, "index": 84}, {"type": "text", "coordinates": [398, 359, 541, 376], "content": " are isomorphic. Then their", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [69, 374, 188, 390], "content": "simple-current groups ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [188, 376, 214, 388], "content": "\\mathbb{Z}_{r+1}", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [214, 374, 241, 390], "content": " and ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [241, 376, 266, 388], "content": "\\mathbb{Z}_{s+1}", "score": 0.93, "index": 89}, {"type": "text", "coordinates": [266, 374, 542, 390], "content": " must be isomorphic (since simple-currents must get", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [70, 389, 235, 404], "content": "mapped to simple-currents), so ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [235, 394, 263, 399], "content": "r=s", "score": 0.88, "index": 92}, {"type": "text", "coordinates": [263, 389, 412, 404], "content": ". Now compare the numbers ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [412, 390, 440, 403], "content": "\|\|P_{+}\|\|", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [440, 389, 541, 404], "content": "of highest-weights:", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [72, 403, 150, 418], "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "score": 0.93, "index": 96}, {"type": "text", "coordinates": [150, 399, 224, 420], "content": ", which forces ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [225, 405, 258, 414], "content": "m=k", "score": 0.93, "index": 98}, {"type": "text", "coordinates": [258, 399, 264, 420], "content": ".", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [93, 418, 540, 433], "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [70, 432, 541, 448], "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [69, 442, 382, 466], "content": "[18, Table 3]. Here we will simply state that those weights in ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [382, 446, 431, 463], "content": "P_{+}^{k}(X_{r}^{(1)})", "score": 0.94, "index": 103}, {"type": "text", "coordinates": [431, 442, 543, 466], "content": " with second smallest", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [69, 462, 258, 479], "content": "q-dimension are precisely the orbit ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [258, 465, 280, 475], "content": "S\\Lambda_{\\star}", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [281, 462, 349, 479], "content": ", except for: ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [349, 465, 371, 477], "content": "A_{r,1}", "score": 0.84, "index": 108}, {"type": "text", "coordinates": [371, 462, 378, 479], "content": "; ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [378, 465, 401, 477], "content": "B_{r,k}", "score": 0.91, "index": 110}, {"type": "text", "coordinates": [401, 462, 423, 479], "content": " for ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [424, 465, 453, 475], "content": "k\\leq3", "score": 0.91, "index": 112}, {"type": "text", "coordinates": [453, 462, 460, 479], "content": "; ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [461, 465, 536, 477], "content": "C_{2,2},C_{2,3},C_{3,2}", "score": 0.93, "index": 114}, {"type": "text", "coordinates": [537, 462, 541, 479], "content": ";", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [71, 479, 94, 491], "content": "D_{r,k}", "score": 0.92, "index": 116}, {"type": "text", "coordinates": [94, 478, 116, 492], "content": " for ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [116, 479, 145, 489], "content": "k\\leq2", "score": 0.9, "index": 118}, {"type": "text", "coordinates": [145, 478, 151, 492], "content": "; ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [152, 479, 174, 491], "content": "E_{6,k}", "score": 0.92, "index": 120}, {"type": "text", "coordinates": [175, 478, 196, 492], "content": " for ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [196, 479, 226, 489], "content": "k\\leq2", "score": 0.92, "index": 122}, {"type": "text", "coordinates": [226, 478, 255, 492], "content": "; and ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [255, 479, 360, 491], "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "score": 0.93, "index": 124}, {"type": "text", "coordinates": [361, 478, 382, 492], "content": " for ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [383, 479, 412, 489], "content": "k\\leq4", "score": 0.92, "index": 126}, {"type": "text", "coordinates": [412, 478, 416, 492], "content": ".", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [95, 494, 116, 506], "content": "C_{r,k}", "score": 0.93, "index": 128}, {"type": "text", "coordinates": [117, 492, 141, 506], "content": " and ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [141, 494, 167, 506], "content": "B_{s,m}", "score": 0.94, "index": 130}, {"type": "text", "coordinates": [167, 492, 540, 506], "content": " both have two simple-currents, but their fusion rings can\u2019t be isomorphic", "score": 1.0, "index": 131}, {"type": "text", "coordinates": [71, 506, 232, 521], "content": "(generically) because the orbit ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [232, 507, 257, 519], "content": "J^{i}\\Lambda_{1}", "score": 0.93, "index": 133}, {"type": "text", "coordinates": [258, 506, 540, 521], "content": " has the second smallest q-dimension for both algebras", "score": 1.0, "index": 134}, {"type": "text", "coordinates": [70, 522, 277, 535], "content": "at generic rank/level, but the numbers ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [277, 521, 327, 535], "content": "Q_{j}(J^{i}\\Lambda_{1})", "score": 0.95, "index": 136}, {"type": "text", "coordinates": [327, 522, 506, 535], "content": " for the two algebras are different.", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [93, 534, 367, 551], "content": "Another useful invariant involves the set of integers ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [367, 537, 373, 546], "content": "\\ell", "score": 0.85, "index": 139}, {"type": "text", "coordinates": [373, 534, 435, 551], "content": "coprime to ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [435, 537, 453, 546], "content": "\\kappa N", "score": 0.89, "index": 141}, {"type": "text", "coordinates": [454, 534, 509, 551], "content": " for which ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [509, 535, 527, 546], "content": "0^{(\\ell)}", "score": 0.88, "index": 143}, {"type": "text", "coordinates": [528, 534, 542, 551], "content": " is", "score": 1.0, "index": 144}, {"type": "text", "coordinates": [70, 551, 539, 565], "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "score": 1.0, "index": 145}, {"type": "text", "coordinates": [68, 561, 172, 583], "content": "the q-dimension of ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [172, 566, 190, 577], "content": "0^{(\\ell)}", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [191, 561, 199, 583], "content": " (", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [199, 568, 204, 577], "content": "\\ell", "score": 0.76, "index": 149}, {"type": "text", "coordinates": [205, 561, 267, 583], "content": "coprime to ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [268, 569, 281, 577], "content": "2\\kappa", "score": 0.78, "index": 151}, {"type": "text", "coordinates": [281, 561, 374, 583], "content": ") for the algebras ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [375, 564, 453, 579], "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "score": 0.94, "index": 153}, {"type": "text", "coordinates": [453, 561, 539, 583], "content": "is, respectively,", "score": 1.0, "index": 154}, {"type": "interline_equation", "coordinates": [145, 595, 464, 720], "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "score": 0.94, "index": 155}]	[]	[{"type": "block", "coordinates": [145, 595, 464, 720], "content": "", "caption": ""}, {"type": "inline", "coordinates": [135, 120, 178, 133], "content": "\\mathcal{R}(X_{r,k})", "caption": ""}, {"type": "inline", "coordinates": [411, 119, 432, 132], "content": "\\S2.2", "caption": ""}, {"type": "inline", "coordinates": [434, 143, 540, 156], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "caption": ""}, {"type": "inline", "coordinates": [101, 156, 164, 170], "content": "X_{r,k}\\neq Y_{s,m}", "caption": ""}, {"type": "inline", "coordinates": [206, 156, 240, 169], "content": "X_{r},Y_{s}", "caption": ""}, {"type": "inline", "coordinates": [164, 171, 263, 185], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "caption": ""}, {"type": "inline", "coordinates": [303, 171, 321, 184], "content": "r,k", "caption": ""}, {"type": "inline", "coordinates": [383, 171, 483, 185], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "caption": ""}, {"type": "inline", "coordinates": [71, 185, 284, 200], "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "caption": ""}, {"type": "inline", "coordinates": [323, 186, 352, 199], "content": "r\\geq3", "caption": ""}, {"type": "inline", "coordinates": [71, 200, 182, 214], "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "caption": ""}, {"type": "inline", "coordinates": [71, 215, 171, 229], "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "caption": ""}, {"type": "inline", "coordinates": [226, 216, 255, 227], "content": "r\\equiv s", "caption": ""}, {"type": "inline", "coordinates": [71, 230, 171, 244], "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "caption": ""}, {"type": "inline", "coordinates": [197, 229, 296, 244], "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "caption": ""}, {"type": "inline", "coordinates": [71, 245, 169, 259], "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "caption": ""}, {"type": "inline", "coordinates": [177, 244, 275, 259], "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "caption": ""}, {"type": "inline", "coordinates": [305, 244, 405, 259], "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "caption": ""}, {"type": "inline", "coordinates": [190, 269, 291, 284], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "caption": ""}, {"type": "inline", "coordinates": [326, 269, 347, 282], "content": "a\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [366, 267, 381, 282], "content": "\\widetilde{\\Lambda}_{a}", "caption": ""}, {"type": "inline", "coordinates": 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330, 197, 344], "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "caption": ""}, {"type": "inline", "coordinates": [286, 361, 328, 374], "content": "\\mathcal{R}(A_{r,k})", "caption": ""}, {"type": "inline", "coordinates": [353, 361, 398, 374], "content": "\\mathcal{R}(A_{s,m})", "caption": ""}, {"type": "inline", "coordinates": [188, 376, 214, 388], "content": "\\mathbb{Z}_{r+1}", "caption": ""}, {"type": "inline", "coordinates": [241, 376, 266, 388], "content": "\\mathbb{Z}_{s+1}", "caption": ""}, {"type": "inline", "coordinates": [235, 394, 263, 399], "content": "r=s", "caption": ""}, {"type": "inline", "coordinates": [412, 390, 440, 403], "content": "\|\|P_{+}\|\|", "caption": ""}, {"type": "inline", "coordinates": [72, 403, 150, 418], "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "caption": ""}, {"type": "inline", "coordinates": [225, 405, 258, 414], "content": "m=k", "caption": ""}, {"type": "inline", "coordinates": [382, 446, 431, 463], "content": "P_{+}^{k}(X_{r}^{(1)})", "caption": ""}, {"type": "inline", "coordinates": [258, 465, 280, 475], "content": "S\\Lambda_{\\star}", "caption": ""}, {"type": "inline", "coordinates": [349, 465, 371, 477], "content": "A_{r,1}", "caption": ""}, {"type": "inline", "coordinates": [378, 465, 401, 477], "content": "B_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [424, 465, 453, 475], "content": "k\\leq3", "caption": ""}, {"type": "inline", "coordinates": [461, 465, 536, 477], "content": "C_{2,2},C_{2,3},C_{3,2}", "caption": ""}, {"type": "inline", "coordinates": [71, 479, 94, 491], "content": "D_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [116, 479, 145, 489], "content": "k\\leq2", "caption": ""}, {"type": "inline", "coordinates": [152, 479, 174, 491], "content": "E_{6,k}", "caption": ""}, {"type": "inline", "coordinates": [196, 479, 226, 489], "content": "k\\leq2", "caption": ""}, {"type": "inline", "coordinates": [255, 479, 360, 491], "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "caption": ""}, {"type": "inline", "coordinates": [383, 479, 412, 489], "content": "k\\leq4", "caption": ""}, {"type": "inline", "coordinates": [95, 494, 116, 506], "content": "C_{r,k}", "caption": ""}, {"type": "inline", "coordinates": [141, 494, 167, 506], "content": "B_{s,m}", "caption": ""}, {"type": "inline", "coordinates": [232, 507, 257, 519], "content": "J^{i}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [277, 521, 327, 535], "content": "Q_{j}(J^{i}\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [367, 537, 373, 546], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [435, 537, 453, 546], "content": "\\kappa N", "caption": ""}, {"type": "inline", "coordinates": [509, 535, 527, 546], "content": "0^{(\\ell)}", "caption": ""}, {"type": "inline", "coordinates": [172, 566, 190, 577], "content": "0^{(\\ell)}", "caption": ""}, {"type": "inline", "coordinates": [199, 568, 204, 577], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [268, 569, 281, 577], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [375, 564, 453, 579], "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "5. Affine fusion ring isomorphisms ", "text_level": 1, "page_idx": 20}, {"type": "text", "text": "We conclude the paper with the determination of all isomorphisms among the affine fusion rings $\\mathcal{R}(X_{r,k})$ . Recall Definition 2.1 and the discussion in $\\S2.2$ . ", "page_idx": 20}, {"type": "text", "text": "Theorem 5.1. The complete list of fusion ring isomorphisms $\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})$ when $X_{r,k}\\neq Y_{s,m}$ (where $X_{r},Y_{s}$ are simple) is: rank-level duality $\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$ for all $r,k$ , as well as $\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$ ; $\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})$ for all $r\\geq3$ ; $\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})$ ; $\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})$ whenever $r\\equiv s$ (mod 2); $\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})$ and $\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})$ ; $\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})$ , $\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})$ , and $\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})$ . ", "page_idx": 20}, {"type": "text", "text": "The isomorphism $\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$ takes $a\\Lambda_{1}$ to $\\widetilde{\\Lambda}_{a}$ . The isomorphism $\\mathcal{R}(F_{4,2})\\cong$ $\\mathcal{R}(E_{8,3})$ was found in [14]; it relates $\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}$ , $2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}$ , $\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}$ . The isomorphism $\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})$ was found i n [34,14]; a corresponde nce which works is $\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}$ , $2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}$ , $\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}$ , $\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}$ , $3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}$ , and $\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}$ . ", "page_idx": 20}, {"type": "text", "text": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $\\mathcal{R}(A_{r,k})$ and $\\mathcal{R}(A_{s,m})$ are isomorphic. Then their simple-current groups $\\mathbb{Z}_{r+1}$ and $\\mathbb{Z}_{s+1}$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $r=s$ . Now compare the numbers $\|\|P_{+}\|\|$ of highest-weights: $\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)$ , which forces $m=k$ . ", "page_idx": 20}, {"type": "text", "text": "It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $P_{+}^{k}(X_{r}^{(1)})$ with second smallest q-dimension are precisely the orbit $S\\Lambda_{\\star}$ , except for: $A_{r,1}$ ; $B_{r,k}$ for $k\\leq3$ ; $C_{2,2},C_{2,3},C_{3,2}$ ; $D_{r,k}$ for $k\\leq2$ ; $E_{6,k}$ for $k\\leq2$ ; and $E_{7,k},E_{8,k},F_{4,k},G_{2,k}$ for $k\\leq4$ . ", "page_idx": 20}, {"type": "text", "text": "$C_{r,k}$ and $B_{s,m}$ both have two simple-currents, but their fusion rings can\u2019t be isomorphic (generically) because the orbit $J^{i}\\Lambda_{1}$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $Q_{j}(J^{i}\\Lambda_{1})$ for the two algebras are different. ", "page_idx": 20}, {"type": "text", "text": "Another useful invariant involves the set of integers $\\ell$ coprime to $\\kappa N$ for which $0^{(\\ell)}$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $0^{(\\ell)}$ ( $\\ell$ coprime to $2\\kappa$ ) for the algebras $B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$ is, respectively, ", "page_idx": 20}, {"type": "equation", "text": "$$\n\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi 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371.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [497.0, 331.0, 1141.0, 331.0, 1141.0, 371.0, 497.0, 371.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1203.0, 331.0, 1211.0, 331.0, 1211.0, 371.0, 1203.0, 371.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [829.0, 2033.0, 870.0, 2033.0, 870.0, 2070.0, 829.0, 2070.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [561.0, 207.0, 1138.0, 207.0, 1138.0, 246.0, 561.0, 246.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "title", "bbox": [200, 71, 410, 86], "lines": [{"bbox": [201, 74, 409, 88], "spans": [{"bbox": [201, 74, 409, 88], "score": 1.0, "content": "5. Affine fusion ring isomorphisms", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [70, 102, 542, 132], "lines": [{"bbox": [94, 103, 541, 120], "spans": [{"bbox": [94, 103, 541, 120], "score": 1.0, "content": "We conclude the paper with the determination of all isomorphisms among the affine", "type": "text"}], "index": 1}, {"bbox": [72, 119, 435, 133], "spans": [{"bbox": [72, 119, 135, 133], "score": 1.0, "content": "fusion rings ", "type": "text"}, {"bbox": [135, 120, 178, 133], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [178, 119, 410, 133], "score": 1.0, "content": ". Recall Definition 2.1 and the discussion in ", "type": "text"}, {"bbox": [411, 119, 432, 132], "score": 0.3, "content": "\\S2.2", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [433, 119, 435, 133], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "text", "bbox": [70, 138, 541, 259], "lines": [{"bbox": [94, 140, 540, 159], "spans": [{"bbox": [94, 140, 434, 159], "score": 1.0, "content": "Theorem 5.1. The complete list of fusion ring isomorphisms ", "type": "text"}, {"bbox": [434, 143, 540, 156], "score": 0.92, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 106}], "index": 3}, {"bbox": [72, 156, 321, 172], "spans": [{"bbox": [72, 156, 101, 172], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [101, 156, 164, 170], "score": 0.92, "content": "X_{r,k}\\neq Y_{s,m}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [164, 156, 206, 172], "score": 1.0, "content": " (where ", "type": "text"}, {"bbox": [206, 156, 240, 169], "score": 0.9, "content": "X_{r},Y_{s}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [240, 156, 321, 172], "score": 1.0, "content": " are simple) is:", "type": "text"}], "index": 4}, {"bbox": [70, 169, 489, 187], "spans": [{"bbox": [70, 169, 164, 187], "score": 1.0, "content": "rank-level duality ", "type": "text"}, {"bbox": [164, 171, 263, 185], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 99}, {"bbox": [263, 169, 302, 187], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [303, 171, 321, 184], "score": 0.88, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [321, 169, 382, 187], "score": 1.0, "content": ", as well as ", "type": "text"}, {"bbox": [383, 171, 483, 185], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [484, 169, 489, 187], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [71, 185, 358, 203], "spans": [{"bbox": [71, 185, 284, 200], "score": 0.86, "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [284, 185, 323, 203], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [323, 186, 352, 199], "score": 0.89, "content": "r\\geq3", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [353, 185, 358, 203], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [71, 200, 188, 217], "spans": [{"bbox": [71, 200, 182, 214], "score": 0.89, "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "type": "inline_equation", "height": 14, "width": 111}, {"bbox": [182, 200, 188, 217], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [71, 214, 307, 231], "spans": [{"bbox": [71, 215, 171, 229], "score": 0.9, "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 214, 226, 231], "score": 1.0, "content": " whenever ", "type": "text"}, {"bbox": [226, 216, 255, 227], "score": 0.48, "content": "r\\equiv s", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [256, 214, 307, 231], "score": 1.0, "content": " (mod 2);", "type": "text"}], "index": 8}, {"bbox": [71, 229, 302, 245], "spans": [{"bbox": [71, 230, 171, 244], "score": 0.89, "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 230, 197, 245], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [197, 229, 296, 244], "score": 0.87, "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "type": "inline_equation", "height": 15, "width": 99}, {"bbox": [296, 230, 302, 245], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [71, 244, 408, 261], "spans": [{"bbox": [71, 245, 169, 259], "score": 0.89, "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [170, 244, 176, 261], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [177, 244, 275, 259], "score": 0.85, "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [275, 244, 304, 261], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [305, 244, 405, 259], "score": 0.9, "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 15, "width": 100}, {"bbox": [405, 244, 408, 261], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 6.5}, {"type": "text", "bbox": [70, 266, 541, 343], "lines": [{"bbox": [93, 267, 541, 285], "spans": [{"bbox": [93, 267, 190, 285], "score": 1.0, "content": "The isomorphism ", "type": "text"}, {"bbox": [190, 269, 291, 284], "score": 0.91, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [292, 267, 326, 285], "score": 1.0, "content": " takes ", "type": "text"}, {"bbox": [326, 269, 347, 282], "score": 0.86, "content": "a\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [348, 267, 365, 285], "score": 1.0, "content": " to", "type": "text"}, {"bbox": [366, 267, 381, 282], "score": 0.9, "content": "\\widetilde{\\Lambda}_{a}", "type": "inline_equation", "height": 15, "width": 15}, {"bbox": [381, 267, 485, 285], "score": 1.0, "content": ". The isomorphism ", "type": "text"}, {"bbox": [486, 270, 541, 284], "score": 0.89, "content": "\\mathcal{R}(F_{4,2})\\cong", "type": "inline_equation", "height": 14, "width": 55}], "index": 11}, {"bbox": [71, 283, 540, 300], "spans": [{"bbox": [71, 288, 113, 300], "score": 0.93, "content": "\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [113, 285, 272, 300], "score": 1.0, "content": " was found in [14]; it relates ", "type": "text"}, {"bbox": [272, 283, 323, 299], "score": 0.89, "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [323, 285, 331, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [331, 284, 388, 299], "score": 0.88, "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [389, 285, 396, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [397, 285, 447, 299], "score": 0.93, "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [448, 285, 455, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [456, 285, 506, 299], "score": 0.93, "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [507, 285, 540, 300], "score": 1.0, "content": ". The", "type": "text"}], "index": 12}, {"bbox": [70, 299, 542, 315], "spans": [{"bbox": [70, 299, 142, 315], "score": 1.0, "content": "isomorphism ", "type": "text"}, {"bbox": [142, 300, 244, 314], "score": 0.91, "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [244, 299, 542, 315], "score": 1.0, "content": " was found i n [34,14]; a corresponde nce which works is", "type": "text"}], "index": 13}, {"bbox": [71, 314, 540, 331], "spans": [{"bbox": [71, 315, 118, 330], "score": 0.92, "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 15, "width": 47}, {"bbox": [118, 315, 124, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [124, 315, 177, 329], "score": 0.88, "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [177, 315, 183, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [184, 315, 236, 330], "score": 0.87, "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [236, 315, 242, 331], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [243, 314, 302, 329], "score": 0.9, "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [302, 315, 308, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [308, 315, 417, 330], "score": 0.89, "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 109}, {"bbox": [418, 315, 424, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [424, 315, 477, 330], "score": 0.9, "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [477, 315, 483, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [484, 315, 536, 329], "score": 0.93, "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [536, 315, 540, 331], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [70, 329, 201, 345], "spans": [{"bbox": [70, 329, 94, 345], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 330, 197, 344], "score": 0.91, "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [198, 329, 201, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13}, {"type": "text", "bbox": [70, 344, 541, 416], "lines": [{"bbox": [95, 345, 540, 361], "spans": [{"bbox": [95, 345, 540, 361], "score": 1.0, "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "type": "text"}], "index": 16}, {"bbox": [69, 359, 541, 376], "spans": [{"bbox": [69, 359, 286, 376], "score": 1.0, "content": "rings, case by case. For example, suppose ", "type": "text"}, {"bbox": [286, 361, 328, 374], "score": 0.94, "content": "\\mathcal{R}(A_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [328, 359, 353, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [353, 361, 398, 374], "score": 0.94, "content": "\\mathcal{R}(A_{s,m})", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [398, 359, 541, 376], "score": 1.0, "content": " are isomorphic. Then their", "type": "text"}], "index": 17}, {"bbox": [69, 374, 542, 390], "spans": [{"bbox": [69, 374, 188, 390], "score": 1.0, "content": "simple-current groups ", "type": "text"}, {"bbox": [188, 376, 214, 388], "score": 0.92, "content": "\\mathbb{Z}_{r+1}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [214, 374, 241, 390], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 376, 266, 388], "score": 0.93, "content": "\\mathbb{Z}_{s+1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [266, 374, 542, 390], "score": 1.0, "content": " must be isomorphic (since simple-currents must get", "type": "text"}], "index": 18}, {"bbox": [70, 389, 541, 404], "spans": [{"bbox": [70, 389, 235, 404], "score": 1.0, "content": "mapped to simple-currents), so ", "type": "text"}, {"bbox": [235, 394, 263, 399], "score": 0.88, "content": "r=s", "type": "inline_equation", "height": 5, "width": 28}, {"bbox": [263, 389, 412, 404], "score": 1.0, "content": ". Now compare the numbers ", "type": "text"}, {"bbox": [412, 390, 440, 403], "score": 0.94, "content": "\|\|P_{+}\|\|", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [440, 389, 541, 404], "score": 1.0, "content": "of highest-weights:", "type": "text"}], "index": 19}, {"bbox": [72, 399, 264, 420], "spans": [{"bbox": [72, 403, 150, 418], "score": 0.93, "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [150, 399, 224, 420], "score": 1.0, "content": ", which forces ", "type": "text"}, {"bbox": [225, 405, 258, 414], "score": 0.93, "content": "m=k", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [258, 399, 264, 420], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18}, {"type": "text", "bbox": [70, 416, 541, 490], "lines": [{"bbox": [93, 418, 540, 433], "spans": [{"bbox": [93, 418, 540, 433], "score": 1.0, "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "type": "text"}], "index": 21}, {"bbox": [70, 432, 541, 448], "spans": [{"bbox": [70, 432, 541, 448], "score": 1.0, "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "type": "text"}], "index": 22}, {"bbox": [69, 442, 543, 466], "spans": [{"bbox": [69, 442, 382, 466], "score": 1.0, "content": "[18, Table 3]. Here we will simply state that those weights in ", "type": "text"}, {"bbox": [382, 446, 431, 463], "score": 0.94, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [431, 442, 543, 466], "score": 1.0, "content": " with second smallest", "type": "text"}], "index": 23}, {"bbox": [69, 462, 541, 479], "spans": [{"bbox": [69, 462, 258, 479], "score": 1.0, "content": "q-dimension are precisely the orbit ", "type": "text"}, {"bbox": [258, 465, 280, 475], "score": 0.92, "content": "S\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [281, 462, 349, 479], "score": 1.0, "content": ", except for: ", "type": "text"}, {"bbox": [349, 465, 371, 477], "score": 0.84, "content": "A_{r,1}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [371, 462, 378, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [378, 465, 401, 477], "score": 0.91, "content": "B_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [401, 462, 423, 479], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [424, 465, 453, 475], "score": 0.91, "content": "k\\leq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [453, 462, 460, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [461, 465, 536, 477], "score": 0.93, "content": "C_{2,2},C_{2,3},C_{3,2}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [537, 462, 541, 479], "score": 1.0, "content": ";", "type": "text"}], "index": 24}, {"bbox": [71, 478, 416, 492], "spans": [{"bbox": [71, 479, 94, 491], "score": 0.92, "content": "D_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [94, 478, 116, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [116, 479, 145, 489], "score": 0.9, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [145, 478, 151, 492], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [152, 479, 174, 491], "score": 0.92, "content": "E_{6,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [175, 478, 196, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [196, 479, 226, 489], "score": 0.92, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [226, 478, 255, 492], "score": 1.0, "content": "; and ", "type": "text"}, {"bbox": [255, 479, 360, 491], "score": 0.93, "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "type": "inline_equation", "height": 12, "width": 105}, {"bbox": [361, 478, 382, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [383, 479, 412, 489], "score": 0.92, "content": "k\\leq4", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [412, 478, 416, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 23}, {"type": "text", "bbox": [71, 490, 541, 533], "lines": [{"bbox": [95, 492, 540, 506], "spans": [{"bbox": [95, 494, 116, 506], "score": 0.93, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [117, 492, 141, 506], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [141, 494, 167, 506], "score": 0.94, "content": "B_{s,m}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [167, 492, 540, 506], "score": 1.0, "content": " both have two simple-currents, but their fusion rings can\u2019t be isomorphic", "type": "text"}], "index": 26}, {"bbox": [71, 506, 540, 521], "spans": [{"bbox": [71, 506, 232, 521], "score": 1.0, "content": "(generically) because the orbit ", "type": "text"}, {"bbox": [232, 507, 257, 519], "score": 0.93, "content": "J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [258, 506, 540, 521], "score": 1.0, "content": " has the second smallest q-dimension for both algebras", "type": "text"}], "index": 27}, {"bbox": [70, 521, 506, 535], "spans": [{"bbox": [70, 522, 277, 535], "score": 1.0, "content": "at generic rank/level, but the numbers ", "type": "text"}, {"bbox": [277, 521, 327, 535], "score": 0.95, "content": "Q_{j}(J^{i}\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [327, 522, 506, 535], "score": 1.0, "content": " for the two algebras are different.", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [70, 534, 541, 579], "lines": [{"bbox": [93, 534, 542, 551], "spans": [{"bbox": [93, 534, 367, 551], "score": 1.0, "content": "Another useful invariant involves the set of integers ", "type": "text"}, {"bbox": [367, 537, 373, 546], "score": 0.85, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [373, 534, 435, 551], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [435, 537, 453, 546], "score": 0.89, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [454, 534, 509, 551], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [509, 535, 527, 546], "score": 0.88, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [528, 534, 542, 551], "score": 1.0, "content": " is", "type": "text"}], "index": 29}, {"bbox": [70, 551, 539, 565], "spans": [{"bbox": [70, 551, 539, 565], "score": 1.0, "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "type": "text"}], "index": 30}, {"bbox": [68, 561, 539, 583], "spans": [{"bbox": [68, 561, 172, 583], "score": 1.0, "content": "the q-dimension of ", "type": "text"}, {"bbox": [172, 566, 190, 577], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [191, 561, 199, 583], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [199, 568, 204, 577], "score": 0.76, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [205, 561, 267, 583], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [268, 569, 281, 577], "score": 0.78, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [281, 561, 374, 583], "score": 1.0, "content": ") for the algebras ", "type": "text"}, {"bbox": [375, 564, 453, 579], "score": 0.94, "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [453, 561, 539, 583], "score": 1.0, "content": "is, respectively,", "type": "text"}], "index": 31}], "index": 30}, {"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 20, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [{"type": "discarded", "bbox": [298, 731, 311, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "21", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [200, 71, 410, 86], "lines": [{"bbox": [201, 74, 409, 88], "spans": [{"bbox": [201, 74, 409, 88], "score": 1.0, "content": "5. Affine fusion ring isomorphisms", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_20", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 102, 542, 132], "lines": [{"bbox": [94, 103, 541, 120], "spans": [{"bbox": [94, 103, 541, 120], "score": 1.0, "content": "We conclude the paper with the determination of all isomorphisms among the affine", "type": "text"}], "index": 1}, {"bbox": [72, 119, 435, 133], "spans": [{"bbox": [72, 119, 135, 133], "score": 1.0, "content": "fusion rings ", "type": "text"}, {"bbox": [135, 120, 178, 133], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [178, 119, 410, 133], "score": 1.0, "content": ". Recall Definition 2.1 and the discussion in ", "type": "text"}, {"bbox": [411, 119, 432, 132], "score": 0.3, "content": "\\S2.2", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [433, 119, 435, 133], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1.5, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [72, 103, 541, 133]}, {"type": "text", "bbox": [70, 138, 541, 259], "lines": [{"bbox": [94, 140, 540, 159], "spans": [{"bbox": [94, 140, 434, 159], "score": 1.0, "content": "Theorem 5.1. The complete list of fusion ring isomorphisms ", "type": "text"}, {"bbox": [434, 143, 540, 156], "score": 0.92, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 106}], "index": 3}, {"bbox": [72, 156, 321, 172], "spans": [{"bbox": [72, 156, 101, 172], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [101, 156, 164, 170], "score": 0.92, "content": "X_{r,k}\\neq Y_{s,m}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [164, 156, 206, 172], "score": 1.0, "content": " (where ", "type": "text"}, {"bbox": [206, 156, 240, 169], "score": 0.9, "content": "X_{r},Y_{s}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [240, 156, 321, 172], "score": 1.0, "content": " are simple) is:", "type": "text"}], "index": 4}, {"bbox": [70, 169, 489, 187], "spans": [{"bbox": [70, 169, 164, 187], "score": 1.0, "content": "rank-level duality ", "type": "text"}, {"bbox": [164, 171, 263, 185], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 99}, {"bbox": [263, 169, 302, 187], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [303, 171, 321, 184], "score": 0.88, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [321, 169, 382, 187], "score": 1.0, "content": ", as well as ", "type": "text"}, {"bbox": [383, 171, 483, 185], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [484, 169, 489, 187], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [71, 185, 358, 203], "spans": [{"bbox": [71, 185, 284, 200], "score": 0.86, "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [284, 185, 323, 203], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [323, 186, 352, 199], "score": 0.89, "content": "r\\geq3", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [353, 185, 358, 203], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [71, 200, 188, 217], "spans": [{"bbox": [71, 200, 182, 214], "score": 0.89, "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "type": "inline_equation", "height": 14, "width": 111}, {"bbox": [182, 200, 188, 217], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [71, 214, 307, 231], "spans": [{"bbox": [71, 215, 171, 229], "score": 0.9, "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 214, 226, 231], "score": 1.0, "content": " whenever ", "type": "text"}, {"bbox": [226, 216, 255, 227], "score": 0.48, "content": "r\\equiv s", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [256, 214, 307, 231], "score": 1.0, "content": " (mod 2);", "type": "text"}], "index": 8}, {"bbox": [71, 229, 302, 245], "spans": [{"bbox": [71, 230, 171, 244], "score": 0.89, "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 230, 197, 245], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [197, 229, 296, 244], "score": 0.87, "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "type": "inline_equation", "height": 15, "width": 99}, {"bbox": [296, 230, 302, 245], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [71, 244, 408, 261], "spans": [{"bbox": [71, 245, 169, 259], "score": 0.89, "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [170, 244, 176, 261], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [177, 244, 275, 259], "score": 0.85, "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [275, 244, 304, 261], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [305, 244, 405, 259], "score": 0.9, "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 15, "width": 100}, {"bbox": [405, 244, 408, 261], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 6.5, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [70, 140, 540, 261]}, {"type": "text", "bbox": [70, 266, 541, 343], "lines": [{"bbox": [93, 267, 541, 285], "spans": [{"bbox": [93, 267, 190, 285], "score": 1.0, "content": "The isomorphism ", "type": "text"}, {"bbox": [190, 269, 291, 284], "score": 0.91, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [292, 267, 326, 285], "score": 1.0, "content": " takes ", "type": "text"}, {"bbox": [326, 269, 347, 282], "score": 0.86, "content": "a\\Lambda_{1}", 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The isomorphism ", "type": "text"}, {"bbox": [486, 270, 541, 284], "score": 0.89, "content": "\\mathcal{R}(F_{4,2})\\cong", "type": "inline_equation", "height": 14, "width": 55}], "index": 11}, {"bbox": [71, 283, 540, 300], "spans": [{"bbox": [71, 288, 113, 300], "score": 0.93, "content": "\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [113, 285, 272, 300], "score": 1.0, "content": " was found in [14]; it relates ", "type": "text"}, {"bbox": [272, 283, 323, 299], "score": 0.89, "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [323, 285, 331, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [331, 284, 388, 299], "score": 0.88, "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [389, 285, 396, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [397, 285, 447, 299], "score": 0.93, "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [448, 285, 455, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [456, 285, 506, 299], "score": 0.93, "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [507, 285, 540, 300], "score": 1.0, "content": ". The", "type": "text"}], "index": 12}, {"bbox": [70, 299, 542, 315], "spans": [{"bbox": [70, 299, 142, 315], "score": 1.0, "content": "isomorphism ", "type": "text"}, {"bbox": [142, 300, 244, 314], "score": 0.91, "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [244, 299, 542, 315], "score": 1.0, "content": " was found i n [34,14]; a corresponde nce which works is", "type": "text"}], "index": 13}, {"bbox": [71, 314, 540, 331], "spans": [{"bbox": [71, 315, 118, 330], "score": 0.92, "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 15, "width": 47}, {"bbox": [118, 315, 124, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [124, 315, 177, 329], "score": 0.88, "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [177, 315, 183, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [184, 315, 236, 330], "score": 0.87, "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [236, 315, 242, 331], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [243, 314, 302, 329], "score": 0.9, "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [302, 315, 308, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [308, 315, 417, 330], "score": 0.89, "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 109}, {"bbox": [418, 315, 424, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [424, 315, 477, 330], "score": 0.9, "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [477, 315, 483, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [484, 315, 536, 329], "score": 0.93, "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [536, 315, 540, 331], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [70, 329, 201, 345], "spans": [{"bbox": [70, 329, 94, 345], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 330, 197, 344], "score": 0.91, "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [198, 329, 201, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [70, 267, 542, 345]}, {"type": "text", "bbox": [70, 344, 541, 416], "lines": [{"bbox": [95, 345, 540, 361], "spans": [{"bbox": [95, 345, 540, 361], "score": 1.0, "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "type": "text"}], "index": 16}, {"bbox": [69, 359, 541, 376], "spans": [{"bbox": [69, 359, 286, 376], "score": 1.0, "content": "rings, case by case. For example, suppose ", "type": "text"}, {"bbox": [286, 361, 328, 374], "score": 0.94, "content": "\\mathcal{R}(A_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [328, 359, 353, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [353, 361, 398, 374], "score": 0.94, "content": "\\mathcal{R}(A_{s,m})", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [398, 359, 541, 376], "score": 1.0, "content": " are isomorphic. Then their", "type": "text"}], "index": 17}, {"bbox": [69, 374, 542, 390], "spans": [{"bbox": [69, 374, 188, 390], "score": 1.0, "content": "simple-current groups ", "type": "text"}, {"bbox": [188, 376, 214, 388], "score": 0.92, "content": "\\mathbb{Z}_{r+1}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [214, 374, 241, 390], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 376, 266, 388], "score": 0.93, "content": "\\mathbb{Z}_{s+1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [266, 374, 542, 390], "score": 1.0, "content": " must be isomorphic (since simple-currents must get", "type": "text"}], "index": 18}, {"bbox": [70, 389, 541, 404], "spans": [{"bbox": [70, 389, 235, 404], "score": 1.0, "content": "mapped to simple-currents), so ", "type": "text"}, {"bbox": [235, 394, 263, 399], "score": 0.88, "content": "r=s", "type": "inline_equation", "height": 5, "width": 28}, {"bbox": [263, 389, 412, 404], "score": 1.0, "content": ". Now compare the numbers ", "type": "text"}, {"bbox": [412, 390, 440, 403], "score": 0.94, "content": "\|\|P_{+}\|\|", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [440, 389, 541, 404], "score": 1.0, "content": "of highest-weights:", "type": "text"}], "index": 19}, {"bbox": [72, 399, 264, 420], "spans": [{"bbox": [72, 403, 150, 418], "score": 0.93, "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [150, 399, 224, 420], "score": 1.0, "content": ", which forces ", "type": "text"}, {"bbox": [225, 405, 258, 414], "score": 0.93, "content": "m=k", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [258, 399, 264, 420], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [69, 345, 542, 420]}, {"type": "text", "bbox": [70, 416, 541, 490], "lines": [{"bbox": [93, 418, 540, 433], "spans": [{"bbox": [93, 418, 540, 433], "score": 1.0, "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "type": "text"}], "index": 21}, {"bbox": [70, 432, 541, 448], "spans": [{"bbox": [70, 432, 541, 448], "score": 1.0, "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "type": "text"}], "index": 22}, {"bbox": [69, 442, 543, 466], "spans": [{"bbox": [69, 442, 382, 466], "score": 1.0, "content": "[18, Table 3]. Here we will simply state that those weights in ", "type": "text"}, {"bbox": [382, 446, 431, 463], "score": 0.94, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [431, 442, 543, 466], "score": 1.0, "content": " with second smallest", "type": "text"}], "index": 23}, {"bbox": [69, 462, 541, 479], "spans": [{"bbox": [69, 462, 258, 479], "score": 1.0, "content": "q-dimension are precisely the orbit ", "type": "text"}, {"bbox": [258, 465, 280, 475], "score": 0.92, "content": "S\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [281, 462, 349, 479], "score": 1.0, "content": ", except for: ", "type": "text"}, {"bbox": [349, 465, 371, 477], "score": 0.84, "content": "A_{r,1}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [371, 462, 378, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [378, 465, 401, 477], "score": 0.91, "content": "B_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [401, 462, 423, 479], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [424, 465, 453, 475], "score": 0.91, "content": "k\\leq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [453, 462, 460, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [461, 465, 536, 477], "score": 0.93, "content": "C_{2,2},C_{2,3},C_{3,2}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [537, 462, 541, 479], "score": 1.0, "content": ";", "type": "text"}], "index": 24}, {"bbox": [71, 478, 416, 492], "spans": [{"bbox": [71, 479, 94, 491], "score": 0.92, "content": "D_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [94, 478, 116, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [116, 479, 145, 489], "score": 0.9, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [145, 478, 151, 492], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [152, 479, 174, 491], "score": 0.92, "content": "E_{6,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [175, 478, 196, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [196, 479, 226, 489], "score": 0.92, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [226, 478, 255, 492], "score": 1.0, "content": "; and ", "type": "text"}, {"bbox": [255, 479, 360, 491], "score": 0.93, "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "type": "inline_equation", "height": 12, "width": 105}, {"bbox": [361, 478, 382, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [383, 479, 412, 489], "score": 0.92, "content": "k\\leq4", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [412, 478, 416, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 23, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [69, 418, 543, 492]}, {"type": "text", "bbox": [71, 490, 541, 533], "lines": [{"bbox": [95, 492, 540, 506], "spans": [{"bbox": [95, 494, 116, 506], "score": 0.93, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [117, 492, 141, 506], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [141, 494, 167, 506], "score": 0.94, "content": "B_{s,m}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [167, 492, 540, 506], "score": 1.0, "content": " both have two simple-currents, but their fusion rings can\u2019t be isomorphic", "type": "text"}], "index": 26}, {"bbox": [71, 506, 540, 521], "spans": [{"bbox": [71, 506, 232, 521], "score": 1.0, "content": "(generically) because the orbit ", "type": "text"}, {"bbox": [232, 507, 257, 519], "score": 0.93, "content": "J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [258, 506, 540, 521], "score": 1.0, "content": " has the second smallest q-dimension for both algebras", "type": "text"}], "index": 27}, {"bbox": [70, 521, 506, 535], "spans": [{"bbox": [70, 522, 277, 535], "score": 1.0, "content": "at generic rank/level, but the numbers ", "type": "text"}, {"bbox": [277, 521, 327, 535], "score": 0.95, "content": "Q_{j}(J^{i}\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [327, 522, 506, 535], "score": 1.0, "content": " for the two algebras are different.", "type": "text"}], "index": 28}], "index": 27, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [70, 492, 540, 535]}, {"type": "text", "bbox": [70, 534, 541, 579], "lines": [{"bbox": [93, 534, 542, 551], "spans": [{"bbox": [93, 534, 367, 551], "score": 1.0, "content": "Another useful invariant involves the set of integers ", "type": "text"}, {"bbox": [367, 537, 373, 546], "score": 0.85, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [373, 534, 435, 551], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [435, 537, 453, 546], "score": 0.89, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [454, 534, 509, 551], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [509, 535, 527, 546], "score": 0.88, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [528, 534, 542, 551], "score": 1.0, "content": " is", "type": "text"}], "index": 29}, {"bbox": [70, 551, 539, 565], "spans": [{"bbox": [70, 551, 539, 565], "score": 1.0, "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "type": "text"}], "index": 30}, {"bbox": [68, 561, 539, 583], "spans": [{"bbox": [68, 561, 172, 583], "score": 1.0, "content": "the q-dimension of ", "type": "text"}, {"bbox": [172, 566, 190, 577], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [191, 561, 199, 583], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [199, 568, 204, 577], "score": 0.76, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [205, 561, 267, 583], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [268, 569, 281, 577], "score": 0.78, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [281, 561, 374, 583], "score": 1.0, "content": ") for the algebras ", "type": "text"}, {"bbox": [375, 564, 453, 579], "score": 0.94, "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [453, 561, 539, 583], "score": 1.0, "content": "is, respectively,", "type": "text"}], "index": 31}], "index": 30, "page_num": "page_20", "page_size": [612.0, 792.0], "bbox_fs": [68, 534, 542, 583]}, {"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32, "page_num": "page_20", "page_size": [612.0, 792.0]}]}
0002044v1	19	Next, note that we know from $$\Lambda_{1}$$ × $$\Lambda_{1}$$ that $$\pi\Lambda_{2}$$ is $$\Lambda_{2}$$ or $$2\Lambda_{1}$$ . As in $$\S4.2$$ , the fusion $$(2\Lambda_{1})$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k{-}2$$ times) contains the simple-current $$J_{v}0$$ , but $$\Lambda_{2}$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k-2$$ times) doesn’t, and thus $$\pi\Lambda_{2}=\Lambda_{2}$$ . Assume $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ . Using the fusions $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ (for $$1<\ell<r-2$$ ), and noting that equals 0 only when $$\ell\,=\,r+1\,-\,k/2$$ , we see that $$\pi\Lambda_{\ell+1}\;=\;\Lambda_{\ell+1}$$ except possibly for $$\ell=r+1-k/2$$ (hence $$2r-2\geq k\geq4)$$ ). For that $$\ell$$ , use q-dimensions: which is valid for these $$k$$ . So we also know $$\pi\Lambda_{i}=\Lambda_{i}$$ for all $$i\le r-2$$ , and we are done. All that remains is $$D_{r,2}$$ . Recall the $$\lambda^{i}$$ defined in $$\S3.4$$ . Note that $$\mathcal{D}(\Lambda_{r})\;=\;\sqrt{r}$$ , $$\mathcal{D}(\lambda^{a})=2$$ , and $$S_{\lambda^{a}\lambda^{b}}/S_{0\lambda^{b}}=2\cos(\pi a b/r)$$ . For $$r\neq4$$ , the q-dimensions force $$\pi\Lambda_{1}=\lambda^{m}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,\lambda^{m^{\prime}}$$ , and $$S_{\Lambda_{1}\Lambda_{1}}\,=\,S_{\lambda^{m}\lambda^{m^{\prime}}}$$ says $$m m^{\prime}\,\equiv\,\pm1$$ (mod $$2r$$ ). So without loss of generality we may take $$m=m^{\prime}=1$$ . The rest of the argument is easy. For $$D_{4,2}$$ , we can force $$\pi\Lambda_{1}=\Lambda_{1}$$ , and then eliminate $$\pi^{\prime}\Lambda_{1}=\Lambda_{r-1}$$ or $$\Lambda_{r}$$ by $$S_{\Lambda_{1}\Lambda_{1}}\ne$$ $$0=S_{\Lambda_{1}\Lambda_{r}}=S_{\Lambda_{1}\Lambda_{r-1}}$$ . The rest of the argument is as before. 4.6. The arguments for the exceptional algebras The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym- metries) given in §§3.5-3.9. For example, consider $$E_{6}^{(1)}$$ for $$k\geq2$$ . Proposition 4.1 tells us $$\pi\Lambda_{1}=C^{a}J^{b}\Lambda_{1}$$ for some $$a,b$$ , and we know $$\pi^{\prime}J0\,=\,J^{c}0$$ for $$c=\pm1$$ . Hence from (2.7b) we get $$k b\not\equiv-1$$ (mod 3). Hitting $$\pi$$ with $$\pi[-b]^{-1}C^{a}$$ , we need consider only $$\pi\Lambda_{1}=\Lambda_{1}$$ . It is now immediate that $$\pi\Lambda_{5}=\Lambda_{5}$$ , by commuting $$\pi$$ with $$C$$ . From (3.6a) we get that $$\pi$$ must permute $$\Lambda_{2}$$ and $$2\Lambda_{1}$$ . Compare (3.6c) with (3.6d): since for any $$k\geq2$$ they have different numbers of summands, we find in fact that $$\pi$$ will fix both $$\Lambda_{2}$$ (hence $$\Lambda_{4}$$ ) and $$2\Lambda_{1}$$ . From (3.6b) we get that $$\pi$$ permutes $$\Lambda_{6}$$ and $$\Lambda_{1}+\Lambda_{5}$$ , and so (3.6d) now tells us $$\pi\Lambda_{6}=\Lambda_{6}$$ . Finally, (3.6c) implies (for $$k\geq3$$ ) $$\pi\Lambda_{3}=\Lambda_{3}$$ (since $$C\pi=\pi C$$ ), and we are done for $$k\geq3$$ . Since $$\{\Lambda_{1},\Lambda_{2},\Lambda_{4},\Lambda_{5},\Lambda_{6}\}$$ is a fusion-generator for $$k=2$$ (see $$\S2.2)$$ , we are also done for $$k=2$$ . For $${E}_{8}^{(1)}$$ when $$k\geq7$$ , (3.7a) tells us that $$\Lambda_{2},\Lambda_{7},2\Lambda_{1}$$ are permuted. For those $$k$$ , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so $$\Lambda_{2},\Lambda_{7},2\Lambda_{1}$$ must all be fixed. The fusion product (3.7c) also tells us that $$\Lambda_{3},\Lambda_{6},\Lambda_{8},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}$$ are permuted; (3.7d) then says that the sets $$\{\Lambda_{6},\Lambda_{8}\}$$ , $$\{\Lambda_{3},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}\}$$ , and $$\{2\Lambda_{2},\Lambda_{2}+\Lambda_{7},3\Lambda_{1},2\Lambda_{1}+$$ $$\Lambda_{2},2\Lambda_{1}+\Lambda_{7},4\Lambda_{1}\right\}$$ are stabilised. Now (3.7b) implies $$\Lambda_{3},\Lambda_{6},\Lambda_{8},2\Lambda_{7}$$ are all fixed, while the set $$\{\Lambda_{4},\Lambda_{1}+\Lambda_{3}\}$$ is stabilised. Comparing $$(3.7\mathrm{e})$$ and (3.7f), we get that $$\Lambda_{4}$$ is fixed and $$\Lambda_{5},\Lambda_{7}+\Lambda_{8}$$ are permuted. Finally, $$\left(3.7\mathrm{g}\right)$$ shows $$\Lambda_{5}$$ also is fixed. To do $${E}_{8}^{(1)}$$ when $$k\leq6$$ , knowing q-dimensions really simplifies things.	<p>Next, note that we know from $$\Lambda_{1}$$ × $$\Lambda_{1}$$ that $$\pi\Lambda_{2}$$ is $$\Lambda_{2}$$ or $$2\Lambda_{1}$$ . As in $$\S4.2$$ , the fusion $$(2\Lambda_{1})$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k{-}2$$ times) contains the simple-current $$J_{v}0$$ , but $$\Lambda_{2}$$ × $$\Lambda_{1}$$ × · · · × $$\Lambda_{1}$$ ( $$k-2$$ times) doesn’t, and thus $$\pi\Lambda_{2}=\Lambda_{2}$$ .</p> <p>Assume $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ . Using the fusions $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ (for $$1<\ell<r-2$$ ), and noting that</p> <p>equals 0 only when $$\ell\,=\,r+1\,-\,k/2$$ , we see that $$\pi\Lambda_{\ell+1}\;=\;\Lambda_{\ell+1}$$ except possibly for $$\ell=r+1-k/2$$ (hence $$2r-2\geq k\geq4)$$ ). For that $$\ell$$ , use q-dimensions:</p> <p>which is valid for these $$k$$ . So we also know $$\pi\Lambda_{i}=\Lambda_{i}$$ for all $$i\le r-2$$ , and we are done.</p> <p>All that remains is $$D_{r,2}$$ . Recall the $$\lambda^{i}$$ defined in $$\S3.4$$ . Note that $$\mathcal{D}(\Lambda_{r})\;=\;\sqrt{r}$$ , $$\mathcal{D}(\lambda^{a})=2$$ , and $$S_{\lambda^{a}\lambda^{b}}/S_{0\lambda^{b}}=2\cos(\pi a b/r)$$ . For $$r\neq4$$ , the q-dimensions force $$\pi\Lambda_{1}=\lambda^{m}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,\lambda^{m^{\prime}}$$ , and $$S_{\Lambda_{1}\Lambda_{1}}\,=\,S_{\lambda^{m}\lambda^{m^{\prime}}}$$ says $$m m^{\prime}\,\equiv\,\pm1$$ (mod $$2r$$ ). So without loss of generality we may take $$m=m^{\prime}=1$$ . The rest of the argument is easy.</p> <p>For $$D_{4,2}$$ , we can force $$\pi\Lambda_{1}=\Lambda_{1}$$ , and then eliminate $$\pi^{\prime}\Lambda_{1}=\Lambda_{r-1}$$ or $$\Lambda_{r}$$ by $$S_{\Lambda_{1}\Lambda_{1}}\ne$$ $$0=S_{\Lambda_{1}\Lambda_{r}}=S_{\Lambda_{1}\Lambda_{r-1}}$$ . The rest of the argument is as before.</p> <p>4.6. The arguments for the exceptional algebras</p> <p>The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym- metries) given in §§3.5-3.9.</p> <p>For example, consider $$E_{6}^{(1)}$$ for $$k\geq2$$ . Proposition 4.1 tells us $$\pi\Lambda_{1}=C^{a}J^{b}\Lambda_{1}$$ for some $$a,b$$ , and we know $$\pi^{\prime}J0\,=\,J^{c}0$$ for $$c=\pm1$$ . Hence from (2.7b) we get $$k b\not\equiv-1$$ (mod 3). Hitting $$\pi$$ with $$\pi[-b]^{-1}C^{a}$$ , we need consider only $$\pi\Lambda_{1}=\Lambda_{1}$$ . It is now immediate that $$\pi\Lambda_{5}=\Lambda_{5}$$ , by commuting $$\pi$$ with $$C$$ . From (3.6a) we get that $$\pi$$ must permute $$\Lambda_{2}$$ and $$2\Lambda_{1}$$ . Compare (3.6c) with (3.6d): since for any $$k\geq2$$ they have different numbers of summands, we find in fact that $$\pi$$ will fix both $$\Lambda_{2}$$ (hence $$\Lambda_{4}$$ ) and $$2\Lambda_{1}$$ . From (3.6b) we get that $$\pi$$ permutes $$\Lambda_{6}$$ and $$\Lambda_{1}+\Lambda_{5}$$ , and so (3.6d) now tells us $$\pi\Lambda_{6}=\Lambda_{6}$$ . Finally, (3.6c) implies (for $$k\geq3$$ ) $$\pi\Lambda_{3}=\Lambda_{3}$$ (since $$C\pi=\pi C$$ ), and we are done for $$k\geq3$$ . Since $$\{\Lambda_{1},\Lambda_{2},\Lambda_{4},\Lambda_{5},\Lambda_{6}\}$$ is a fusion-generator for $$k=2$$ (see $$\S2.2)$$ , we are also done for $$k=2$$ .</p> <p>For $${E}_{8}^{(1)}$$ when $$k\geq7$$ , (3.7a) tells us that $$\Lambda_{2},\Lambda_{7},2\Lambda_{1}$$ are permuted. For those $$k$$ , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so $$\Lambda_{2},\Lambda_{7},2\Lambda_{1}$$ must all be fixed. The fusion product (3.7c) also tells us that $$\Lambda_{3},\Lambda_{6},\Lambda_{8},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}$$ are permuted; (3.7d) then says that the sets $$\{\Lambda_{6},\Lambda_{8}\}$$ , $$\{\Lambda_{3},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}\}$$ , and $$\{2\Lambda_{2},\Lambda_{2}+\Lambda_{7},3\Lambda_{1},2\Lambda_{1}+$$ $$\Lambda_{2},2\Lambda_{1}+\Lambda_{7},4\Lambda_{1}\right\}$$ are stabilised. Now (3.7b) implies $$\Lambda_{3},\Lambda_{6},\Lambda_{8},2\Lambda_{7}$$ are all fixed, while the set $$\{\Lambda_{4},\Lambda_{1}+\Lambda_{3}\}$$ is stabilised. Comparing $$(3.7\mathrm{e})$$ and (3.7f), we get that $$\Lambda_{4}$$ is fixed and $$\Lambda_{5},\Lambda_{7}+\Lambda_{8}$$ are permuted. Finally, $$\left(3.7\mathrm{g}\right)$$ shows $$\Lambda_{5}$$ also is fixed. To do $${E}_{8}^{(1)}$$ when $$k\leq6$$ , knowing q-dimensions really simplifies things.</p>	[{"type": "text", "coordinates": [71, 70, 569, 114], "content": "Next, note that we know from $$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{1}$$ that $$\\pi\\Lambda_{2}$$ is $$\\Lambda_{2}$$ or $$2\\Lambda_{1}$$ . As in $$\\S4.2$$ , the fusion\n$$(2\\Lambda_{1})$$ \u00d7 $$\\Lambda_{1}$$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $$\\Lambda_{1}$$ ( $$k{-}2$$ times) contains the simple-current $$J_{v}0$$ , but $$\\Lambda_{2}$$ \u00d7 $$\\Lambda_{1}$$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $$\\Lambda_{1}$$\n( $$k-2$$ times) doesn\u2019t, and thus $$\\pi\\Lambda_{2}=\\Lambda_{2}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [93, 114, 527, 129], "content": "Assume $$\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$$ . Using the fusions $$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{\\ell}$$ (for $$1<\\ell<r-2$$ ), and noting that", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [124, 142, 488, 171], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [69, 182, 541, 211], "content": "equals 0 only when $$\\ell\\,=\\,r+1\\,-\\,k/2$$ , we see that $$\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}$$ except possibly for\n$$\\ell=r+1-k/2$$ (hence $$2r-2\\geq k\\geq4)$$ ). For that $$\\ell$$ , use q-dimensions:", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [205, 225, 405, 257], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [71, 267, 531, 282], "content": "which is valid for these $$k$$ . So we also know $$\\pi\\Lambda_{i}=\\Lambda_{i}$$ for all $$i\\le r-2$$ , and we are done.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 284, 542, 342], "content": "All that remains is $$D_{r,2}$$ . Recall the $$\\lambda^{i}$$ defined in $$\\S3.4$$ . Note that $$\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}$$ ,\n$$\\mathcal{D}(\\lambda^{a})=2$$ , and $$S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)$$ . For $$r\\neq4$$ , the q-dimensions force $$\\pi\\Lambda_{1}=\\lambda^{m}$$\nand $$\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}$$ , and $$S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}$$ says $$m m^{\\prime}\\,\\equiv\\,\\pm1$$ (mod $$2r$$ ). So without loss of\ngenerality we may take $$m=m^{\\prime}=1$$ . The rest of the argument is easy.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 343, 541, 372], "content": "For $$D_{4,2}$$ , we can force $$\\pi\\Lambda_{1}=\\Lambda_{1}$$ , and then eliminate $$\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}$$ or $$\\Lambda_{r}$$ by $$S_{\\Lambda_{1}\\Lambda_{1}}\\ne$$\n$$0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}$$ . The rest of the argument is as before.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [72, 385, 321, 399], "content": "4.6. The arguments for the exceptional algebras", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 406, 540, 436], "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-\nmetries) given in \u00a7\u00a73.5-3.9.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [70, 437, 541, 567], "content": "For example, consider $$E_{6}^{(1)}$$ for $$k\\geq2$$ . Proposition 4.1 tells us $$\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}$$ for some\n$$a,b$$ , and we know $$\\pi^{\\prime}J0\\,=\\,J^{c}0$$ for $$c=\\pm1$$ . Hence from (2.7b) we get $$k b\\not\\equiv-1$$ (mod 3).\nHitting $$\\pi$$ with $$\\pi[-b]^{-1}C^{a}$$ , we need consider only $$\\pi\\Lambda_{1}=\\Lambda_{1}$$ . It is now immediate that\n$$\\pi\\Lambda_{5}=\\Lambda_{5}$$ , by commuting $$\\pi$$ with $$C$$ . From (3.6a) we get that $$\\pi$$ must permute $$\\Lambda_{2}$$ and $$2\\Lambda_{1}$$ .\nCompare (3.6c) with (3.6d): since for any $$k\\geq2$$ they have different numbers of summands,\nwe find in fact that $$\\pi$$ will fix both $$\\Lambda_{2}$$ (hence $$\\Lambda_{4}$$ ) and $$2\\Lambda_{1}$$ . From (3.6b) we get that $$\\pi$$\npermutes $$\\Lambda_{6}$$ and $$\\Lambda_{1}+\\Lambda_{5}$$ , and so (3.6d) now tells us $$\\pi\\Lambda_{6}=\\Lambda_{6}$$ . Finally, (3.6c) implies (for\n$$k\\geq3$$ ) $$\\pi\\Lambda_{3}=\\Lambda_{3}$$ (since $$C\\pi=\\pi C$$ ), and we are done for $$k\\geq3$$ . Since $$\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}$$\nis a fusion-generator for $$k=2$$ (see $$\\S2.2)$$ , we are also done for $$k=2$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 569, 542, 687], "content": "For $${E}_{8}^{(1)}$$ when $$k\\geq7$$ , (3.7a) tells us that $$\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$$ are permuted. For those $$k$$ , the\nhighest multiplicities in (3.7b)\u2013(3.7d) are 3, 1, 2, respectively, so $$\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$$ must all be\nfixed. The fusion product (3.7c) also tells us that $$\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}$$ are permuted;\n(3.7d) then says that the sets $$\\{\\Lambda_{6},\\Lambda_{8}\\}$$ , $$\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}$$ , and $$\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+$$\n$$\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}$$ are stabilised. Now (3.7b) implies $$\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}$$ are all fixed, while\nthe set $$\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}$$ is stabilised. Comparing $$(3.7\\mathrm{e})$$ and (3.7f), we get that $$\\Lambda_{4}$$ is fixed\nand $$\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}$$ are permuted. Finally, $$\\left(3.7\\mathrm{g}\\right)$$ shows $$\\Lambda_{5}$$ also is fixed. To do $${E}_{8}^{(1)}$$ when\n$$k\\leq6$$ , knowing q-dimensions really simplifies things.", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [93, 72, 255, 89], "content": "Next, note that we know from", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [256, 73, 271, 86], "content": "\\Lambda_{1}", "score": 0.85, "index": 2}, {"type": "text", "coordinates": [271, 72, 287, 89], "content": " \u00d7 ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [288, 73, 302, 86], "content": "\\Lambda_{1}", "score": 0.84, "index": 4}, {"type": "text", "coordinates": [303, 72, 331, 89], "content": " that ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [331, 73, 353, 86], "content": "\\pi\\Lambda_{2}", "score": 0.9, "index": 6}, {"type": "text", "coordinates": [353, 72, 368, 89], "content": " is ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [368, 74, 382, 86], "content": "\\Lambda_{2}", "score": 0.88, "index": 8}, {"type": "text", "coordinates": [383, 72, 400, 89], "content": " or ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [400, 74, 420, 86], "content": "2\\Lambda_{1}", "score": 0.87, "index": 10}, {"type": "text", "coordinates": [420, 72, 459, 89], "content": ". As in ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [460, 73, 482, 87], "content": "\\S4.2", "score": 0.6, "index": 12}, {"type": "text", "coordinates": [482, 72, 541, 89], "content": ", the fusion", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [71, 87, 100, 101], "content": "(2\\Lambda_{1})", "score": 0.84, "index": 14}, {"type": "text", "coordinates": [101, 86, 117, 104], "content": " \u00d7 ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [118, 87, 132, 100], "content": "\\Lambda_{1}", "score": 0.87, "index": 16}, {"type": "text", "coordinates": [133, 86, 185, 104], "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [185, 87, 200, 101], "content": "\\Lambda_{1}", "score": 0.85, "index": 18}, {"type": "text", "coordinates": [201, 86, 207, 104], "content": " (", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [207, 88, 230, 100], "content": "k{-}2", "score": 0.53, "index": 20}, {"type": "text", "coordinates": [230, 86, 409, 104], "content": " times) contains the simple-current ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [409, 90, 428, 100], "content": "J_{v}0", "score": 0.91, "index": 22}, {"type": "text", "coordinates": [429, 86, 454, 104], "content": ", but", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [455, 88, 469, 100], "content": "\\Lambda_{2}", "score": 0.88, "index": 24}, {"type": "text", "coordinates": [469, 86, 485, 104], "content": " \u00d7", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [486, 87, 501, 100], "content": "\\Lambda_{1}", "score": 0.87, "index": 26}, {"type": "text", "coordinates": [501, 86, 553, 104], "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [554, 87, 569, 101], "content": "\\Lambda_{1}", "score": 0.86, "index": 28}, {"type": "text", "coordinates": [71, 101, 75, 118], "content": "(", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [75, 102, 104, 114], "content": "k-2", "score": 0.68, "index": 30}, {"type": "text", "coordinates": [104, 101, 237, 118], "content": " times) doesn\u2019t, and thus ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [238, 102, 289, 115], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [289, 101, 294, 118], "content": ".", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [94, 115, 138, 134], "content": "Assume", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [139, 117, 189, 129], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "score": 0.9, "index": 35}, {"type": "text", "coordinates": [189, 115, 290, 134], "content": ". Using the fusions ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [290, 116, 304, 129], "content": "\\Lambda_{1}", "score": 0.83, "index": 37}, {"type": "text", "coordinates": [305, 115, 321, 134], "content": " \u00d7", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [322, 116, 336, 129], "content": "\\Lambda_{\\ell}", "score": 0.77, "index": 39}, {"type": "text", "coordinates": [336, 115, 362, 134], "content": "(for ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [362, 117, 433, 129], "content": "1<\\ell<r-2", "score": 0.85, "index": 41}, {"type": "text", "coordinates": [433, 115, 528, 134], "content": "), and noting that", "score": 1.0, "index": 42}, {"type": "interline_equation", "coordinates": [124, 142, 488, 171], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "score": 0.92, "index": 43}, {"type": "text", "coordinates": [70, 183, 181, 201], "content": "equals 0 only when ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [182, 184, 275, 198], "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [275, 183, 352, 201], "content": ", we see that ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [352, 184, 432, 198], "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "score": 0.91, "index": 47}, {"type": "text", "coordinates": [432, 183, 542, 201], "content": " except possibly for", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [71, 200, 152, 212], "content": "\\ell=r+1-k/2", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [152, 199, 193, 213], "content": " (hence ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [193, 199, 273, 211], "content": "2r-2\\geq k\\geq4)", "score": 0.85, "index": 51}, {"type": "text", "coordinates": [273, 199, 330, 213], "content": "). For that ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [330, 199, 337, 209], "content": "\\ell", "score": 0.81, "index": 53}, {"type": "text", "coordinates": [337, 199, 436, 213], "content": ", use q-dimensions:", "score": 1.0, "index": 54}, {"type": "interline_equation", "coordinates": [205, 225, 405, 257], "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [70, 270, 195, 285], "content": "which is valid for these ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [195, 271, 202, 280], "content": "k", "score": 0.85, "index": 57}, {"type": "text", "coordinates": [203, 270, 299, 285], "content": ". So we also know ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [300, 271, 348, 282], "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "score": 0.92, "index": 59}, {"type": "text", "coordinates": [348, 270, 386, 285], "content": " for all ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [386, 270, 434, 282], "content": "i\\le r-2", "score": 0.88, "index": 61}, {"type": "text", "coordinates": [434, 270, 529, 285], "content": ", and we are done.", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [94, 286, 203, 302], "content": "All that remains is ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [204, 289, 226, 301], "content": "D_{r,2}", "score": 0.91, "index": 64}, {"type": "text", "coordinates": [226, 286, 299, 302], "content": ". Recall the ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [299, 288, 310, 298], "content": "\\lambda^{i}", "score": 0.89, "index": 66}, {"type": "text", "coordinates": [311, 286, 374, 302], "content": " defined in ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [374, 287, 396, 300], "content": "\\S3.4", "score": 0.3, "index": 68}, {"type": "text", "coordinates": [396, 286, 466, 302], "content": ". Note that ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [466, 288, 537, 301], "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [537, 286, 540, 302], "content": ",", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [71, 303, 126, 315], "content": "\\mathcal{D}(\\lambda^{a})=2", "score": 0.93, "index": 72}, {"type": "text", "coordinates": [127, 299, 157, 318], "content": ", and ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [157, 302, 296, 315], "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [296, 299, 326, 318], "content": ". For ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [327, 303, 356, 315], "content": "r\\neq4", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [356, 299, 484, 318], "content": ", the q-dimensions force ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [485, 303, 539, 314], "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "score": 0.89, "index": 78}, {"type": "text", "coordinates": [68, 311, 95, 335], "content": "and ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [96, 315, 160, 329], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [160, 311, 193, 335], "content": ", and ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [193, 318, 281, 330], "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [281, 311, 314, 335], "content": " says ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [315, 317, 375, 327], "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "score": 0.89, "index": 84}, {"type": "text", "coordinates": [375, 311, 412, 335], "content": " (mod ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [412, 316, 425, 327], "content": "2r", "score": 0.56, "index": 86}, {"type": "text", "coordinates": [425, 311, 545, 335], "content": "). So without loss of", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [70, 329, 195, 346], "content": "generality we may take ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [195, 331, 258, 341], "content": "m=m^{\\prime}=1", "score": 0.91, "index": 89}, {"type": "text", "coordinates": [259, 329, 442, 346], "content": ". The rest of the argument is easy.", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [92, 343, 117, 361], "content": "For ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [117, 347, 140, 359], "content": "D_{4,2}", "score": 0.92, "index": 92}, {"type": "text", "coordinates": [140, 343, 215, 361], "content": ", we can force ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [216, 347, 267, 357], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [267, 343, 376, 361], "content": ", and then eliminate ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [376, 345, 443, 358], "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "score": 0.91, "index": 96}, {"type": "text", "coordinates": [443, 343, 461, 361], "content": " or ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [461, 345, 475, 357], "content": "\\Lambda_{r}", "score": 0.89, "index": 98}, {"type": "text", "coordinates": [476, 343, 496, 361], "content": " by ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [496, 345, 542, 358], "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "score": 0.89, "index": 100}, {"type": "inline_equation", "coordinates": [71, 361, 181, 373], "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [181, 358, 389, 375], "content": ". The rest of the argument is as before.", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [71, 387, 322, 402], "content": "4.6. The arguments for the exceptional algebras", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [94, 408, 540, 425], "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [71, 424, 213, 439], "content": "metries) given in \u00a7\u00a73.5-3.9.", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [91, 435, 211, 459], "content": "For example, consider ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [212, 438, 234, 454], "content": "E_{6}^{(1)}", "score": 0.94, "index": 107}, {"type": "text", "coordinates": [234, 435, 254, 459], "content": "for", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [255, 442, 284, 452], "content": "k\\geq2", "score": 0.91, "index": 109}, {"type": "text", "coordinates": [284, 435, 414, 459], "content": ". Proposition 4.1 tells us ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [414, 440, 492, 452], "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "score": 0.94, "index": 111}, {"type": "text", "coordinates": [492, 435, 544, 459], "content": " for some", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [71, 456, 89, 467], "content": "a,b", "score": 0.88, "index": 113}, {"type": "text", "coordinates": [89, 454, 170, 469], "content": ", and we know ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [170, 455, 232, 466], "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "score": 0.93, "index": 115}, {"type": "text", "coordinates": [232, 454, 254, 469], "content": " for ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [255, 457, 293, 466], "content": "c=\\pm1", "score": 0.91, "index": 117}, {"type": "text", "coordinates": [294, 454, 443, 469], "content": ". Hence from (2.7b) we get ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [443, 456, 489, 468], "content": "k b\\not\\equiv-1", "score": 0.9, "index": 119}, {"type": "text", "coordinates": [489, 454, 541, 469], "content": " (mod 3).", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [71, 469, 113, 483], "content": "Hitting ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [114, 474, 121, 480], "content": "\\pi", "score": 0.77, "index": 122}, {"type": "text", "coordinates": [121, 469, 153, 483], "content": " with ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [154, 469, 210, 482], "content": "\\pi[-b]^{-1}C^{a}", "score": 0.94, "index": 124}, {"type": "text", "coordinates": [211, 469, 340, 483], "content": ", we need consider only ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [340, 470, 394, 481], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "score": 0.93, "index": 126}, {"type": "text", "coordinates": [394, 469, 542, 483], "content": ". It is now immediate that", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [71, 485, 122, 496], "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "score": 0.92, "index": 128}, {"type": "text", "coordinates": [122, 484, 206, 498], "content": ", by commuting ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [207, 488, 214, 494], "content": "\\pi", "score": 0.88, "index": 130}, {"type": "text", "coordinates": [214, 484, 244, 498], "content": " with ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [244, 485, 254, 494], "content": "C", "score": 0.9, "index": 132}, {"type": "text", "coordinates": [254, 484, 389, 498], "content": ". From (3.6a) we get that ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [389, 488, 397, 494], "content": "\\pi", "score": 0.87, "index": 134}, {"type": "text", "coordinates": [397, 484, 476, 498], "content": " must permute ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [476, 485, 490, 496], "content": "\\Lambda_{2}", "score": 0.91, "index": 136}, {"type": "text", "coordinates": [490, 484, 516, 498], "content": " and ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [516, 485, 536, 496], "content": "2\\Lambda_{1}", "score": 0.9, "index": 138}, {"type": "text", "coordinates": [536, 484, 540, 498], "content": ".", "score": 1.0, "index": 139}, {"type": "text", "coordinates": [71, 498, 288, 512], "content": "Compare (3.6c) with (3.6d): since for any", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [289, 499, 318, 510], "content": "k\\geq2", "score": 0.92, "index": 141}, {"type": "text", "coordinates": [318, 498, 541, 512], "content": " they have different numbers of summands,", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [71, 513, 180, 526], "content": "we find in fact that ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [180, 517, 187, 523], "content": "\\pi", "score": 0.86, "index": 144}, {"type": "text", "coordinates": [188, 513, 262, 526], "content": " will fix both ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [262, 514, 276, 525], "content": "\\Lambda_{2}", "score": 0.89, "index": 146}, {"type": "text", "coordinates": [277, 513, 319, 526], "content": " (hence ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [320, 514, 334, 525], "content": "\\Lambda_{4}", "score": 0.86, "index": 148}, {"type": "text", "coordinates": [334, 513, 366, 526], "content": ") and ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [367, 514, 387, 524], "content": "2\\Lambda_{1}", "score": 0.89, "index": 150}, {"type": "text", "coordinates": [387, 513, 532, 526], "content": ". From (3.6b) we get that ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [532, 517, 539, 523], "content": "\\pi", "score": 0.82, "index": 152}, {"type": "text", "coordinates": [70, 526, 122, 541], "content": "permutes ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [122, 528, 136, 538], "content": "\\Lambda_{6}", "score": 0.92, "index": 154}, {"type": "text", "coordinates": [137, 526, 162, 541], "content": " and ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [162, 528, 203, 539], "content": "\\Lambda_{1}+\\Lambda_{5}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [203, 526, 345, 541], "content": ", and so (3.6d) now tells us ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [345, 528, 396, 539], "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "score": 0.92, "index": 158}, {"type": "text", "coordinates": [396, 526, 541, 541], "content": ". Finally, (3.6c) implies (for", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [71, 542, 101, 553], "content": "k\\geq3", "score": 0.88, "index": 160}, {"type": "text", "coordinates": [101, 540, 108, 555], "content": ") ", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [109, 542, 160, 553], "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "score": 0.92, "index": 162}, {"type": "text", "coordinates": [160, 540, 198, 555], "content": " (since ", "score": 1.0, "index": 163}, {"type": "inline_equation", "coordinates": [198, 542, 248, 551], "content": "C\\pi=\\pi C", "score": 0.88, "index": 164}, {"type": "text", "coordinates": [249, 540, 367, 555], "content": "), and we are done for", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [368, 542, 398, 553], "content": "k\\geq3", "score": 0.9, "index": 166}, {"type": "text", "coordinates": [398, 540, 437, 555], "content": ". Since ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [438, 542, 540, 554], "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "score": 0.93, "index": 168}, {"type": "text", "coordinates": [69, 555, 199, 569], "content": "is a fusion-generator for ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [200, 557, 228, 565], "content": "k=2", "score": 0.91, "index": 170}, {"type": "text", "coordinates": [229, 555, 255, 569], "content": " (see ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [256, 555, 280, 568], "content": "\\S2.2)", "score": 0.39, "index": 172}, {"type": "text", "coordinates": [280, 555, 397, 569], "content": ", we are also done for ", "score": 1.0, "index": 173}, {"type": "inline_equation", "coordinates": [397, 556, 426, 566], "content": "k=2", "score": 0.9, "index": 174}, {"type": "text", "coordinates": [426, 555, 431, 569], "content": ".", "score": 1.0, "index": 175}, {"type": "text", "coordinates": [89, 564, 117, 589], "content": "For ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [117, 569, 139, 585], "content": "{E}_{8}^{(1)}", "score": 0.94, "index": 177}, {"type": "text", "coordinates": [140, 564, 174, 589], "content": "when ", "score": 1.0, "index": 178}, {"type": "inline_equation", "coordinates": [175, 573, 205, 584], "content": "k\\geq7", "score": 0.91, "index": 179}, {"type": "text", "coordinates": [205, 564, 313, 589], "content": ", (3.7a) tells us that ", "score": 1.0, "index": 180}, {"type": "inline_equation", "coordinates": [313, 571, 372, 585], "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "score": 0.91, "index": 181}, {"type": "text", "coordinates": [372, 564, 508, 589], "content": " are permuted. For those ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [508, 573, 515, 582], "content": "k", "score": 0.82, "index": 183}, {"type": "text", "coordinates": [516, 564, 545, 589], "content": ", the", "score": 1.0, "index": 184}, {"type": "text", "coordinates": [71, 586, 417, 601], "content": "highest multiplicities in (3.7b)\u2013(3.7d) are 3, 1, 2, respectively, so ", "score": 1.0, "index": 185}, {"type": "inline_equation", "coordinates": [417, 586, 475, 599], "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "score": 0.91, "index": 186}, {"type": "text", "coordinates": [476, 586, 541, 601], "content": " must all be", "score": 1.0, "index": 187}, {"type": "text", "coordinates": [70, 600, 336, 615], "content": "fixed. The fusion product (3.7c) also tells us that ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [337, 600, 462, 613], "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "score": 0.92, "index": 189}, {"type": "text", "coordinates": [462, 600, 541, 615], "content": " are permuted;", "score": 1.0, "index": 190}, {"type": "text", "coordinates": [71, 615, 227, 630], "content": "(3.7d) then says that the sets ", "score": 1.0, "index": 191}, {"type": "inline_equation", "coordinates": [227, 615, 272, 628], "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "score": 0.91, "index": 192}, {"type": "text", "coordinates": [273, 615, 279, 630], "content": ", ", "score": 1.0, "index": 193}, {"type": "inline_equation", "coordinates": [279, 615, 376, 628], "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "score": 0.91, "index": 194}, {"type": "text", "coordinates": [377, 615, 406, 630], "content": ", and ", "score": 1.0, "index": 195}, {"type": "inline_equation", "coordinates": [406, 614, 541, 628], "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "score": 0.9, "index": 196}, {"type": "inline_equation", "coordinates": [71, 630, 171, 642], "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "score": 0.9, "index": 197}, {"type": "text", "coordinates": [171, 630, 359, 643], "content": " are stabilised. Now (3.7b) implies ", "score": 1.0, "index": 198}, {"type": "inline_equation", "coordinates": [360, 629, 437, 642], "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "score": 0.91, "index": 199}, {"type": "text", "coordinates": [437, 630, 541, 643], "content": " are all fixed, while", "score": 1.0, "index": 200}, {"type": "text", "coordinates": [70, 644, 111, 659], "content": "the set ", "score": 1.0, "index": 201}, {"type": "inline_equation", "coordinates": [111, 644, 186, 657], "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "score": 0.92, "index": 202}, {"type": "text", "coordinates": [186, 644, 323, 659], "content": " is stabilised. Comparing ", "score": 1.0, "index": 203}, {"type": "inline_equation", "coordinates": [324, 643, 353, 657], "content": "(3.7\\mathrm{e})", "score": 0.25, "index": 204}, {"type": "text", "coordinates": [354, 644, 483, 659], "content": " and (3.7f), we get that ", "score": 1.0, "index": 205}, {"type": "inline_equation", "coordinates": [483, 643, 497, 656], "content": "\\Lambda_{4}", "score": 0.87, "index": 206}, {"type": "text", "coordinates": [498, 644, 541, 659], "content": " is fixed", "score": 1.0, "index": 207}, {"type": "text", "coordinates": [66, 656, 95, 680], "content": "and ", "score": 1.0, "index": 208}, {"type": "inline_equation", "coordinates": [95, 662, 157, 673], "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "score": 0.92, "index": 209}, {"type": "text", "coordinates": [158, 656, 287, 680], "content": " are permuted. Finally, ", "score": 1.0, "index": 210}, {"type": "inline_equation", "coordinates": [288, 660, 318, 673], "content": "\\left(3.7\\mathrm{g}\\right)", "score": 0.49, "index": 211}, {"type": "text", "coordinates": [319, 656, 357, 680], "content": " shows ", "score": 1.0, "index": 212}, {"type": "inline_equation", "coordinates": [357, 660, 371, 672], "content": "\\Lambda_{5}", "score": 0.88, "index": 213}, {"type": "text", "coordinates": [372, 656, 484, 680], "content": " also is fixed. To do ", "score": 1.0, "index": 214}, {"type": "inline_equation", "coordinates": [484, 658, 507, 674], "content": "{E}_{8}^{(1)}", "score": 0.93, "index": 215}, {"type": "text", "coordinates": [508, 656, 540, 680], "content": "when", "score": 1.0, "index": 216}, {"type": "inline_equation", "coordinates": [71, 676, 100, 687], "content": "k\\leq6", "score": 0.91, "index": 217}, {"type": "text", "coordinates": [100, 674, 348, 691], "content": ", knowing q-dimensions really simplifies things.", "score": 1.0, "index": 218}]	[]	[{"type": "block", "coordinates": [124, 142, 488, 171], "content": "", "caption": ""}, {"type": "block", "coordinates": [205, 225, 405, 257], "content": "", "caption": ""}, {"type": "inline", "coordinates": [256, 73, 271, 86], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [288, 73, 302, 86], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [331, 73, 353, 86], "content": "\\pi\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [368, 74, 382, 86], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [400, 74, 420, 86], "content": "2\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [460, 73, 482, 87], "content": "\\S4.2", "caption": ""}, {"type": "inline", "coordinates": [71, 87, 100, 101], "content": "(2\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [118, 87, 132, 100], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [185, 87, 200, 101], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [207, 88, 230, 100], "content": "k{-}2", "caption": ""}, {"type": "inline", "coordinates": [409, 90, 428, 100], "content": "J_{v}0", "caption": ""}, {"type": "inline", "coordinates": [455, 88, 469, 100], "content": "\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [486, 87, 501, 100], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [554, 87, 569, 101], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [75, 102, 104, 114], "content": "k-2", "caption": ""}, {"type": "inline", "coordinates": [238, 102, 289, 115], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [139, 117, 189, 129], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [290, 116, 304, 129], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [322, 116, 336, 129], "content": "\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [362, 117, 433, 129], "content": "1<\\ell<r-2", "caption": ""}, {"type": "inline", "coordinates": [182, 184, 275, 198], "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "caption": ""}, {"type": "inline", "coordinates": [352, 184, 432, 198], "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "caption": ""}, {"type": "inline", "coordinates": [71, 200, 152, 212], "content": "\\ell=r+1-k/2", "caption": ""}, {"type": "inline", "coordinates": [193, 199, 273, 211], "content": "2r-2\\geq k\\geq4)", "caption": ""}, {"type": "inline", "coordinates": [330, 199, 337, 209], "content": "\\ell", "caption": ""}, {"type": "inline", "coordinates": [195, 271, 202, 280], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [300, 271, 348, 282], "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "caption": ""}, {"type": "inline", "coordinates": [386, 270, 434, 282], "content": "i\\le r-2", "caption": ""}, {"type": "inline", "coordinates": [204, 289, 226, 301], "content": "D_{r,2}", "caption": ""}, {"type": "inline", "coordinates": [299, 288, 310, 298], "content": "\\lambda^{i}", "caption": ""}, {"type": "inline", "coordinates": [374, 287, 396, 300], "content": "\\S3.4", "caption": ""}, {"type": "inline", "coordinates": [466, 288, 537, 301], "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "caption": ""}, {"type": "inline", "coordinates": [71, 303, 126, 315], "content": "\\mathcal{D}(\\lambda^{a})=2", "caption": ""}, {"type": "inline", "coordinates": [157, 302, 296, 315], "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "caption": ""}, {"type": "inline", "coordinates": [327, 303, 356, 315], "content": "r\\neq4", "caption": ""}, {"type": "inline", "coordinates": [485, 303, 539, 314], "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "caption": ""}, {"type": "inline", "coordinates": [96, 315, 160, 329], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "caption": ""}, {"type": "inline", "coordinates": [193, 318, 281, 330], "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "caption": ""}, {"type": "inline", "coordinates": [315, 317, 375, 327], "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "caption": ""}, {"type": "inline", "coordinates": [412, 316, 425, 327], "content": "2r", "caption": ""}, {"type": "inline", "coordinates": [195, 331, 258, 341], "content": 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As in $\\S4.2$ , the fusion $(2\\Lambda_{1})$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k{-}2$ times) contains the simple-current $J_{v}0$ , but $\\Lambda_{2}$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k-2$ times) doesn\u2019t, and thus $\\pi\\Lambda_{2}=\\Lambda_{2}$ . ", "page_idx": 19}, {"type": "text", "text": "Assume $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ . Using the fusions $\\Lambda_{1}$ \u00d7 $\\Lambda_{\\ell}$ (for $1<\\ell<r-2$ ), and noting that ", "page_idx": 19}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}\n$$", "text_format": "latex", "page_idx": 19}, {"type": "text", "text": "equals 0 only when $\\ell\\,=\\,r+1\\,-\\,k/2$ , we see that $\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}$ except possibly for $\\ell=r+1-k/2$ (hence $2r-2\\geq k\\geq4)$ ). For that $\\ell$ , use q-dimensions: ", "page_idx": 19}, {"type": "equation", "text": "$$\n\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,\n$$", "text_format": "latex", "page_idx": 19}, {"type": "text", "text": "which is valid for these $k$ . So we also know $\\pi\\Lambda_{i}=\\Lambda_{i}$ for all $i\\le r-2$ , and we are done. ", "page_idx": 19}, {"type": "text", "text": "All that remains is $D_{r,2}$ . Recall the $\\lambda^{i}$ defined in $\\S3.4$ . Note that $\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}$ , $\\mathcal{D}(\\lambda^{a})=2$ , and $S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)$ . For $r\\neq4$ , the q-dimensions force $\\pi\\Lambda_{1}=\\lambda^{m}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}$ , and $S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}$ says $m m^{\\prime}\\,\\equiv\\,\\pm1$ (mod $2r$ ). So without loss of generality we may take $m=m^{\\prime}=1$ . The rest of the argument is easy. ", "page_idx": 19}, {"type": "text", "text": "For $D_{4,2}$ , we can force $\\pi\\Lambda_{1}=\\Lambda_{1}$ , and then eliminate $\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}$ or $\\Lambda_{r}$ by $S_{\\Lambda_{1}\\Lambda_{1}}\\ne$ $0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}$ . The rest of the argument is as before. ", "page_idx": 19}, {"type": "text", "text": "4.6. The arguments for the exceptional algebras ", "page_idx": 19}, {"type": "text", "text": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram symmetries) given in \u00a7\u00a73.5-3.9. ", "page_idx": 19}, {"type": "text", "text": "For example, consider $E_{6}^{(1)}$ for $k\\geq2$ . Proposition 4.1 tells us $\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}$ for some $a,b$ , and we know $\\pi^{\\prime}J0\\,=\\,J^{c}0$ for $c=\\pm1$ . Hence from (2.7b) we get $k b\\not\\equiv-1$ (mod 3). Hitting $\\pi$ with $\\pi[-b]^{-1}C^{a}$ , we need consider only $\\pi\\Lambda_{1}=\\Lambda_{1}$ . It is now immediate that $\\pi\\Lambda_{5}=\\Lambda_{5}$ , by commuting $\\pi$ with $C$ . From (3.6a) we get that $\\pi$ must permute $\\Lambda_{2}$ and $2\\Lambda_{1}$ . Compare (3.6c) with (3.6d): since for any $k\\geq2$ they have different numbers of summands, we find in fact that $\\pi$ will fix both $\\Lambda_{2}$ (hence $\\Lambda_{4}$ ) and $2\\Lambda_{1}$ . From (3.6b) we get that $\\pi$ permutes $\\Lambda_{6}$ and $\\Lambda_{1}+\\Lambda_{5}$ , and so (3.6d) now tells us $\\pi\\Lambda_{6}=\\Lambda_{6}$ . Finally, (3.6c) implies (for $k\\geq3$ ) $\\pi\\Lambda_{3}=\\Lambda_{3}$ (since $C\\pi=\\pi C$ ), and we are done for $k\\geq3$ . Since $\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}$ is a fusion-generator for $k=2$ (see $\\S2.2)$ , we are also done for $k=2$ . ", "page_idx": 19}, {"type": "text", "text": "For ${E}_{8}^{(1)}$ when $k\\geq7$ , (3.7a) tells us that $\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$ are permuted. For those $k$ , the highest multiplicities in (3.7b)\u2013(3.7d) are 3, 1, 2, respectively, so $\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$ must all be fixed. The fusion product (3.7c) also tells us that $\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}$ are permuted; (3.7d) then says that the sets $\\{\\Lambda_{6},\\Lambda_{8}\\}$ , $\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}$ , and $\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+$ $\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}$ are stabilised. Now (3.7b) implies $\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}$ are all fixed, while the set $\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}$ is stabilised. Comparing $(3.7\\mathrm{e})$ and (3.7f), we get that $\\Lambda_{4}$ is fixed and $\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}$ are permuted. Finally, $\\left(3.7\\mathrm{g}\\right)$ shows $\\Lambda_{5}$ also is fixed. To do ${E}_{8}^{(1)}$ when $k\\leq6$ , knowing q-dimensions really simplifies things. 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Using the fusions ", "type": "text"}, {"bbox": [290, 116, 304, 129], "score": 0.83, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [305, 115, 321, 134], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [322, 116, 336, 129], "score": 0.77, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [336, 115, 362, 134], "score": 1.0, "content": "(for ", "type": "text"}, {"bbox": [362, 117, 433, 129], "score": 0.85, "content": "1<\\ell<r-2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [433, 115, 528, 134], "score": 1.0, "content": "), and noting that", "type": "text"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [69, 182, 541, 211], "lines": [{"bbox": [70, 183, 542, 201], "spans": [{"bbox": [70, 183, 181, 201], "score": 1.0, "content": "equals 0 only when ", "type": "text"}, {"bbox": [182, 184, 275, 198], "score": 0.91, "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [275, 183, 352, 201], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [352, 184, 432, 198], "score": 0.91, "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 14, "width": 80}, {"bbox": [432, 183, 542, 201], "score": 1.0, "content": " except possibly for", "type": "text"}], "index": 5}, {"bbox": [71, 199, 436, 213], "spans": [{"bbox": [71, 200, 152, 212], "score": 0.92, "content": "\\ell=r+1-k/2", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [152, 199, 193, 213], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [193, 199, 273, 211], "score": 0.85, "content": "2r-2\\geq k\\geq4)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [273, 199, 330, 213], "score": 1.0, "content": "). For that ", "type": "text"}, {"bbox": [330, 199, 337, 209], "score": 0.81, "content": "\\ell", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [337, 199, 436, 213], "score": 1.0, "content": ", use q-dimensions:", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [71, 267, 531, 282], "lines": [{"bbox": [70, 270, 529, 285], "spans": [{"bbox": [70, 270, 195, 285], "score": 1.0, "content": "which is valid for these ", "type": "text"}, {"bbox": [195, 271, 202, 280], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [203, 270, 299, 285], "score": 1.0, "content": ". So we also know ", "type": "text"}, {"bbox": [300, 271, 348, 282], "score": 0.92, "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [348, 270, 386, 285], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [386, 270, 434, 282], "score": 0.88, "content": "i\\le r-2", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [434, 270, 529, 285], "score": 1.0, "content": ", and we are done.", "type": "text"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [70, 284, 542, 342], "lines": [{"bbox": [94, 286, 540, 302], "spans": [{"bbox": [94, 286, 203, 302], "score": 1.0, "content": "All that remains is ", "type": "text"}, {"bbox": [204, 289, 226, 301], "score": 0.91, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [226, 286, 299, 302], "score": 1.0, "content": ". Recall the ", "type": "text"}, {"bbox": [299, 288, 310, 298], "score": 0.89, "content": "\\lambda^{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [311, 286, 374, 302], "score": 1.0, "content": " defined in ", "type": "text"}, {"bbox": [374, 287, 396, 300], "score": 0.3, "content": "\\S3.4", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [396, 286, 466, 302], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [466, 288, 537, 301], "score": 0.92, "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [537, 286, 540, 302], "score": 1.0, "content": ",", "type": "text"}], "index": 9}, {"bbox": [71, 299, 539, 318], "spans": [{"bbox": [71, 303, 126, 315], "score": 0.93, "content": "\\mathcal{D}(\\lambda^{a})=2", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [127, 299, 157, 318], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [157, 302, 296, 315], "score": 0.91, "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "type": "inline_equation", "height": 13, "width": 139}, {"bbox": [296, 299, 326, 318], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [327, 303, 356, 315], "score": 0.92, "content": "r\\neq4", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [356, 299, 484, 318], "score": 1.0, "content": ", the q-dimensions force ", "type": "text"}, {"bbox": [485, 303, 539, 314], "score": 0.89, "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "type": "inline_equation", "height": 11, "width": 54}], "index": 10}, {"bbox": [68, 311, 545, 335], "spans": [{"bbox": [68, 311, 95, 335], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [96, 315, 160, 329], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [160, 311, 193, 335], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [193, 318, 281, 330], "score": 0.92, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [281, 311, 314, 335], "score": 1.0, "content": " says ", "type": "text"}, {"bbox": [315, 317, 375, 327], "score": 0.89, "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [375, 311, 412, 335], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [412, 316, 425, 327], "score": 0.56, "content": "2r", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [425, 311, 545, 335], "score": 1.0, "content": "). So without loss of", "type": "text"}], "index": 11}, {"bbox": [70, 329, 442, 346], "spans": [{"bbox": [70, 329, 195, 346], "score": 1.0, "content": "generality we may take ", "type": "text"}, {"bbox": [195, 331, 258, 341], "score": 0.91, "content": "m=m^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [259, 329, 442, 346], "score": 1.0, "content": ". The rest of the argument is easy.", "type": "text"}], "index": 12}], "index": 10.5}, {"type": "text", "bbox": [70, 343, 541, 372], "lines": [{"bbox": [92, 343, 542, 361], "spans": [{"bbox": [92, 343, 117, 361], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 347, 140, 359], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 343, 215, 361], "score": 1.0, "content": ", we can force ", "type": "text"}, {"bbox": [216, 347, 267, 357], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [267, 343, 376, 361], "score": 1.0, "content": ", and then eliminate ", "type": "text"}, {"bbox": [376, 345, 443, 358], "score": 0.91, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [443, 343, 461, 361], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [461, 345, 475, 357], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [476, 343, 496, 361], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [496, 345, 542, 358], "score": 0.89, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "type": "inline_equation", "height": 13, "width": 46}], "index": 13}, {"bbox": [71, 358, 389, 375], "spans": [{"bbox": [71, 361, 181, 373], "score": 0.92, "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "type": "inline_equation", "height": 12, "width": 110}, {"bbox": [181, 358, 389, 375], "score": 1.0, "content": ". The rest of the argument is as before.", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [72, 385, 321, 399], "lines": [{"bbox": [71, 387, 322, 402], "spans": [{"bbox": [71, 387, 322, 402], "score": 1.0, "content": "4.6. The arguments for the exceptional algebras", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [70, 406, 540, 436], "lines": [{"bbox": [94, 408, 540, 425], "spans": [{"bbox": [94, 408, 540, 425], "score": 1.0, "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "type": "text"}], "index": 16}, {"bbox": [71, 424, 213, 439], "spans": [{"bbox": [71, 424, 213, 439], "score": 1.0, "content": "metries) given in \u00a7\u00a73.5-3.9.", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [70, 437, 541, 567], "lines": [{"bbox": [91, 435, 544, 459], "spans": [{"bbox": [91, 435, 211, 459], "score": 1.0, "content": "For example, consider ", "type": "text"}, {"bbox": [212, 438, 234, 454], "score": 0.94, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [234, 435, 254, 459], "score": 1.0, "content": "for", "type": "text"}, {"bbox": [255, 442, 284, 452], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [284, 435, 414, 459], "score": 1.0, "content": ". Proposition 4.1 tells us ", "type": "text"}, {"bbox": [414, 440, 492, 452], "score": 0.94, "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [492, 435, 544, 459], "score": 1.0, "content": " for some", "type": "text"}], "index": 18}, {"bbox": [71, 454, 541, 469], "spans": [{"bbox": [71, 456, 89, 467], "score": 0.88, "content": "a,b", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [89, 454, 170, 469], "score": 1.0, "content": ", and we know ", "type": "text"}, {"bbox": [170, 455, 232, 466], "score": 0.93, "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [232, 454, 254, 469], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [255, 457, 293, 466], "score": 0.91, "content": "c=\\pm1", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [294, 454, 443, 469], "score": 1.0, "content": ". Hence from (2.7b) we get ", "type": "text"}, {"bbox": [443, 456, 489, 468], "score": 0.9, "content": "k b\\not\\equiv-1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [489, 454, 541, 469], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}, {"bbox": [71, 469, 542, 483], "spans": [{"bbox": [71, 469, 113, 483], "score": 1.0, "content": "Hitting ", "type": "text"}, {"bbox": [114, 474, 121, 480], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [121, 469, 153, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [154, 469, 210, 482], "score": 0.94, "content": "\\pi[-b]^{-1}C^{a}", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [211, 469, 340, 483], "score": 1.0, "content": ", we need consider only ", "type": "text"}, {"bbox": [340, 470, 394, 481], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [394, 469, 542, 483], "score": 1.0, "content": ". It is now immediate that", "type": "text"}], "index": 20}, {"bbox": [71, 484, 540, 498], "spans": [{"bbox": [71, 485, 122, 496], "score": 0.92, "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [122, 484, 206, 498], "score": 1.0, "content": ", by commuting ", "type": "text"}, {"bbox": [207, 488, 214, 494], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 484, 244, 498], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [244, 485, 254, 494], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [254, 484, 389, 498], "score": 1.0, "content": ". From (3.6a) we get that ", "type": "text"}, {"bbox": [389, 488, 397, 494], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [397, 484, 476, 498], "score": 1.0, "content": " must permute ", "type": "text"}, {"bbox": [476, 485, 490, 496], "score": 0.91, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [490, 484, 516, 498], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [516, 485, 536, 496], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [536, 484, 540, 498], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [71, 498, 541, 512], "spans": [{"bbox": [71, 498, 288, 512], "score": 1.0, "content": "Compare (3.6c) with (3.6d): since for any", "type": "text"}, {"bbox": [289, 499, 318, 510], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [318, 498, 541, 512], "score": 1.0, "content": " they have different numbers of summands,", "type": "text"}], "index": 22}, {"bbox": [71, 513, 539, 526], "spans": [{"bbox": [71, 513, 180, 526], "score": 1.0, "content": "we find in fact that ", "type": "text"}, {"bbox": [180, 517, 187, 523], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [188, 513, 262, 526], "score": 1.0, "content": " will fix both ", "type": "text"}, {"bbox": [262, 514, 276, 525], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [277, 513, 319, 526], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [320, 514, 334, 525], "score": 0.86, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [334, 513, 366, 526], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [367, 514, 387, 524], "score": 0.89, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [387, 513, 532, 526], "score": 1.0, "content": ". From (3.6b) we get that ", "type": "text"}, {"bbox": [532, 517, 539, 523], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}], "index": 23}, {"bbox": [70, 526, 541, 541], "spans": [{"bbox": [70, 526, 122, 541], "score": 1.0, "content": "permutes ", "type": "text"}, {"bbox": [122, 528, 136, 538], "score": 0.92, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [137, 526, 162, 541], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 528, 203, 539], "score": 0.93, "content": "\\Lambda_{1}+\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [203, 526, 345, 541], "score": 1.0, "content": ", and so (3.6d) now tells us ", "type": "text"}, {"bbox": [345, 528, 396, 539], "score": 0.92, "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [396, 526, 541, 541], "score": 1.0, "content": ". Finally, (3.6c) implies (for", "type": "text"}], "index": 24}, {"bbox": [71, 540, 540, 555], "spans": [{"bbox": [71, 542, 101, 553], "score": 0.88, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [101, 540, 108, 555], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [109, 542, 160, 553], "score": 0.92, "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [160, 540, 198, 555], "score": 1.0, "content": " (since ", "type": "text"}, {"bbox": [198, 542, 248, 551], "score": 0.88, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 9, "width": 50}, {"bbox": [249, 540, 367, 555], "score": 1.0, "content": "), and we are done for", "type": "text"}, {"bbox": [368, 542, 398, 553], "score": 0.9, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [398, 540, 437, 555], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [438, 542, 540, 554], "score": 0.93, "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "type": "inline_equation", "height": 12, "width": 102}], "index": 25}, {"bbox": [69, 555, 431, 569], "spans": [{"bbox": [69, 555, 199, 569], "score": 1.0, "content": "is a fusion-generator for ", "type": "text"}, {"bbox": [200, 557, 228, 565], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [229, 555, 255, 569], "score": 1.0, "content": " (see ", "type": "text"}, {"bbox": [256, 555, 280, 568], "score": 0.39, "content": "\\S2.2)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [280, 555, 397, 569], "score": 1.0, "content": ", we are also done for ", "type": "text"}, {"bbox": [397, 556, 426, 566], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [426, 555, 431, 569], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 22}, {"type": "text", "bbox": [70, 569, 542, 687], "lines": [{"bbox": [89, 564, 545, 589], "spans": [{"bbox": [89, 564, 117, 589], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 569, 139, 585], "score": 0.94, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [140, 564, 174, 589], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [175, 573, 205, 584], "score": 0.91, "content": "k\\geq7", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [205, 564, 313, 589], "score": 1.0, "content": ", (3.7a) tells us that ", "type": "text"}, {"bbox": [313, 571, 372, 585], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [372, 564, 508, 589], "score": 1.0, "content": " are permuted. For those ", "type": "text"}, {"bbox": [508, 573, 515, 582], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [516, 564, 545, 589], "score": 1.0, "content": ", the", "type": "text"}], "index": 27}, {"bbox": [71, 586, 541, 601], "spans": [{"bbox": [71, 586, 417, 601], "score": 1.0, "content": "highest multiplicities in (3.7b)\u2013(3.7d) are 3, 1, 2, respectively, so ", "type": "text"}, {"bbox": [417, 586, 475, 599], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [476, 586, 541, 601], "score": 1.0, "content": " must all be", "type": "text"}], "index": 28}, {"bbox": [70, 600, 541, 615], "spans": [{"bbox": [70, 600, 336, 615], "score": 1.0, "content": "fixed. The fusion product (3.7c) also tells us that ", "type": "text"}, {"bbox": [337, 600, 462, 613], "score": 0.92, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [462, 600, 541, 615], "score": 1.0, "content": " are permuted;", "type": "text"}], "index": 29}, {"bbox": [71, 614, 541, 630], "spans": [{"bbox": [71, 615, 227, 630], "score": 1.0, "content": "(3.7d) then says that the sets ", "type": "text"}, {"bbox": [227, 615, 272, 628], "score": 0.91, "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [273, 615, 279, 630], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [279, 615, 376, 628], "score": 0.91, "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [377, 615, 406, 630], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [406, 614, 541, 628], "score": 0.9, "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "type": "inline_equation", "height": 14, "width": 135}], "index": 30}, {"bbox": [71, 629, 541, 643], "spans": [{"bbox": [71, 630, 171, 642], "score": 0.9, "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [171, 630, 359, 643], "score": 1.0, "content": " are stabilised. Now (3.7b) implies ", "type": "text"}, {"bbox": [360, 629, 437, 642], "score": 0.91, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [437, 630, 541, 643], "score": 1.0, "content": " are all fixed, while", "type": "text"}], "index": 31}, {"bbox": [70, 643, 541, 659], "spans": [{"bbox": [70, 644, 111, 659], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [111, 644, 186, 657], "score": 0.92, "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [186, 644, 323, 659], "score": 1.0, "content": " is stabilised. Comparing ", "type": "text"}, {"bbox": [324, 643, 353, 657], "score": 0.25, "content": "(3.7\\mathrm{e})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [354, 644, 483, 659], "score": 1.0, "content": " and (3.7f), we get that ", "type": "text"}, {"bbox": [483, 643, 497, 656], "score": 0.87, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [498, 644, 541, 659], "score": 1.0, "content": " is fixed", "type": "text"}], "index": 32}, {"bbox": [66, 656, 540, 680], "spans": [{"bbox": [66, 656, 95, 680], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 662, 157, 673], "score": 0.92, "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [158, 656, 287, 680], "score": 1.0, "content": " are permuted. Finally, ", "type": "text"}, {"bbox": [288, 660, 318, 673], "score": 0.49, "content": "\\left(3.7\\mathrm{g}\\right)", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [319, 656, 357, 680], "score": 1.0, "content": " shows ", "type": "text"}, {"bbox": [357, 660, 371, 672], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [372, 656, 484, 680], "score": 1.0, "content": " also is fixed. To do ", "type": "text"}, {"bbox": [484, 658, 507, 674], "score": 0.93, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [508, 656, 540, 680], "score": 1.0, "content": "when", "type": "text"}], "index": 33}, {"bbox": [71, 674, 348, 691], "spans": [{"bbox": [71, 676, 100, 687], "score": 0.91, "content": "k\\leq6", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 674, 348, 691], "score": 1.0, "content": ", knowing q-dimensions really simplifies things.", "type": "text"}], "index": 34}], "index": 30.5}], "layout_bboxes": [], "page_idx": 19, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "20", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 569, 114], "lines": [{"bbox": [93, 72, 541, 89], "spans": [{"bbox": [93, 72, 255, 89], "score": 1.0, "content": "Next, note that we know from", "type": "text"}, {"bbox": [256, 73, 271, 86], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [271, 72, 287, 89], "score": 1.0, "content": " \u00d7 ", "type": "text"}, {"bbox": [288, 73, 302, 86], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [303, 72, 331, 89], "score": 1.0, "content": " that ", "type": "text"}, {"bbox": [331, 73, 353, 86], "score": 0.9, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [353, 72, 368, 89], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [368, 74, 382, 86], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [383, 72, 400, 89], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [400, 74, 420, 86], "score": 0.87, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [420, 72, 459, 89], "score": 1.0, "content": ". As in ", "type": "text"}, {"bbox": [460, 73, 482, 87], "score": 0.6, "content": "\\S4.2", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [482, 72, 541, 89], "score": 1.0, "content": ", the fusion", "type": "text"}], "index": 0}, {"bbox": [71, 86, 569, 104], "spans": [{"bbox": [71, 87, 100, 101], "score": 0.84, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [101, 86, 117, 104], "score": 1.0, "content": " \u00d7 ", "type": "text"}, {"bbox": [118, 87, 132, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [133, 86, 185, 104], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "type": "text"}, {"bbox": [185, 87, 200, 101], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [201, 86, 207, 104], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [207, 88, 230, 100], "score": 0.53, "content": "k{-}2", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [230, 86, 409, 104], "score": 1.0, "content": " times) contains the simple-current ", "type": "text"}, {"bbox": [409, 90, 428, 100], "score": 0.91, "content": "J_{v}0", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [429, 86, 454, 104], "score": 1.0, "content": ", but", "type": "text"}, {"bbox": [455, 88, 469, 100], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [469, 86, 485, 104], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [486, 87, 501, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [501, 86, 553, 104], "score": 1.0, "content": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "type": "text"}, {"bbox": [554, 87, 569, 101], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}], "index": 1}, {"bbox": [71, 101, 294, 118], "spans": [{"bbox": [71, 101, 75, 118], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [75, 102, 104, 114], "score": 0.68, "content": "k-2", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [104, 101, 237, 118], "score": 1.0, "content": " times) doesn\u2019t, and thus ", "type": "text"}, {"bbox": [238, 102, 289, 115], "score": 0.92, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [289, 101, 294, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [71, 72, 569, 118]}, {"type": "text", "bbox": [93, 114, 527, 129], "lines": [{"bbox": [94, 115, 528, 134], "spans": [{"bbox": [94, 115, 138, 134], "score": 1.0, "content": "Assume", "type": "text"}, {"bbox": [139, 117, 189, 129], "score": 0.9, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [189, 115, 290, 134], "score": 1.0, "content": ". Using the fusions ", "type": "text"}, {"bbox": [290, 116, 304, 129], "score": 0.83, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [305, 115, 321, 134], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [322, 116, 336, 129], "score": 0.77, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [336, 115, 362, 134], "score": 1.0, "content": "(for ", "type": "text"}, {"bbox": [362, 117, 433, 129], "score": 0.85, "content": "1<\\ell<r-2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [433, 115, 528, 134], "score": 1.0, "content": "), and noting that", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [94, 115, 528, 134]}, {"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_19", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 182, 541, 211], "lines": [{"bbox": [70, 183, 542, 201], "spans": [{"bbox": [70, 183, 181, 201], "score": 1.0, "content": "equals 0 only when ", "type": "text"}, {"bbox": [182, 184, 275, 198], "score": 0.91, "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [275, 183, 352, 201], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [352, 184, 432, 198], "score": 0.91, "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 14, "width": 80}, {"bbox": [432, 183, 542, 201], "score": 1.0, "content": " except possibly for", "type": "text"}], "index": 5}, {"bbox": [71, 199, 436, 213], "spans": [{"bbox": [71, 200, 152, 212], "score": 0.92, "content": "\\ell=r+1-k/2", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [152, 199, 193, 213], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [193, 199, 273, 211], "score": 0.85, "content": "2r-2\\geq k\\geq4)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [273, 199, 330, 213], "score": 1.0, "content": "). For that ", "type": "text"}, {"bbox": [330, 199, 337, 209], "score": 0.81, "content": "\\ell", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [337, 199, 436, 213], "score": 1.0, "content": ", use q-dimensions:", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [70, 183, 542, 213]}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_19", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 267, 531, 282], "lines": [{"bbox": [70, 270, 529, 285], "spans": [{"bbox": [70, 270, 195, 285], "score": 1.0, "content": "which is valid for these ", "type": "text"}, {"bbox": [195, 271, 202, 280], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [203, 270, 299, 285], "score": 1.0, "content": ". So we also know ", "type": "text"}, {"bbox": [300, 271, 348, 282], "score": 0.92, "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [348, 270, 386, 285], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [386, 270, 434, 282], "score": 0.88, "content": "i\\le r-2", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [434, 270, 529, 285], "score": 1.0, "content": ", and we are done.", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [70, 270, 529, 285]}, {"type": "text", "bbox": [70, 284, 542, 342], "lines": [{"bbox": [94, 286, 540, 302], "spans": [{"bbox": [94, 286, 203, 302], "score": 1.0, "content": "All that remains is ", "type": "text"}, {"bbox": [204, 289, 226, 301], "score": 0.91, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [226, 286, 299, 302], "score": 1.0, "content": ". Recall the ", "type": "text"}, {"bbox": [299, 288, 310, 298], "score": 0.89, "content": "\\lambda^{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [311, 286, 374, 302], "score": 1.0, "content": " defined in ", "type": "text"}, {"bbox": [374, 287, 396, 300], "score": 0.3, "content": "\\S3.4", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [396, 286, 466, 302], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [466, 288, 537, 301], "score": 0.92, "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [537, 286, 540, 302], "score": 1.0, "content": ",", "type": "text"}], "index": 9}, {"bbox": [71, 299, 539, 318], "spans": [{"bbox": [71, 303, 126, 315], "score": 0.93, "content": "\\mathcal{D}(\\lambda^{a})=2", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [127, 299, 157, 318], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [157, 302, 296, 315], "score": 0.91, "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "type": "inline_equation", "height": 13, "width": 139}, {"bbox": [296, 299, 326, 318], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [327, 303, 356, 315], "score": 0.92, "content": "r\\neq4", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [356, 299, 484, 318], "score": 1.0, "content": ", the q-dimensions force ", "type": "text"}, {"bbox": [485, 303, 539, 314], "score": 0.89, "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "type": "inline_equation", "height": 11, "width": 54}], "index": 10}, {"bbox": [68, 311, 545, 335], "spans": [{"bbox": [68, 311, 95, 335], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [96, 315, 160, 329], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [160, 311, 193, 335], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [193, 318, 281, 330], "score": 0.92, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [281, 311, 314, 335], "score": 1.0, "content": " says ", "type": "text"}, {"bbox": [315, 317, 375, 327], "score": 0.89, "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [375, 311, 412, 335], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [412, 316, 425, 327], "score": 0.56, "content": "2r", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [425, 311, 545, 335], "score": 1.0, "content": "). So without loss of", "type": "text"}], "index": 11}, {"bbox": [70, 329, 442, 346], "spans": [{"bbox": [70, 329, 195, 346], "score": 1.0, "content": "generality we may take ", "type": "text"}, {"bbox": [195, 331, 258, 341], "score": 0.91, "content": "m=m^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [259, 329, 442, 346], "score": 1.0, "content": ". The rest of the argument is easy.", "type": "text"}], "index": 12}], "index": 10.5, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [68, 286, 545, 346]}, {"type": "text", "bbox": [70, 343, 541, 372], "lines": [{"bbox": [92, 343, 542, 361], "spans": [{"bbox": [92, 343, 117, 361], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 347, 140, 359], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 343, 215, 361], "score": 1.0, "content": ", we can force ", "type": "text"}, {"bbox": [216, 347, 267, 357], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [267, 343, 376, 361], "score": 1.0, "content": ", and then eliminate ", "type": "text"}, {"bbox": [376, 345, 443, 358], "score": 0.91, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [443, 343, 461, 361], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [461, 345, 475, 357], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [476, 343, 496, 361], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [496, 345, 542, 358], "score": 0.89, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "type": "inline_equation", "height": 13, "width": 46}], "index": 13}, {"bbox": [71, 358, 389, 375], "spans": [{"bbox": [71, 361, 181, 373], "score": 0.92, "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "type": "inline_equation", "height": 12, "width": 110}, {"bbox": [181, 358, 389, 375], "score": 1.0, "content": ". The rest of the argument is as before.", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [71, 343, 542, 375]}, {"type": "text", "bbox": [72, 385, 321, 399], "lines": [{"bbox": [71, 387, 322, 402], "spans": [{"bbox": [71, 387, 322, 402], "score": 1.0, "content": "4.6. The arguments for the exceptional algebras", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [71, 387, 322, 402]}, {"type": "text", "bbox": [70, 406, 540, 436], "lines": [{"bbox": [94, 408, 540, 425], "spans": [{"bbox": [94, 408, 540, 425], "score": 1.0, "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "type": "text"}], "index": 16}, {"bbox": [71, 424, 213, 439], "spans": [{"bbox": [71, 424, 213, 439], "score": 1.0, "content": "metries) given in \u00a7\u00a73.5-3.9.", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [71, 408, 540, 439]}, {"type": "text", "bbox": [70, 437, 541, 567], "lines": [{"bbox": [91, 435, 544, 459], "spans": [{"bbox": [91, 435, 211, 459], "score": 1.0, "content": "For example, consider ", "type": "text"}, {"bbox": [212, 438, 234, 454], "score": 0.94, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [234, 435, 254, 459], "score": 1.0, "content": "for", "type": "text"}, {"bbox": [255, 442, 284, 452], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [284, 435, 414, 459], "score": 1.0, "content": ". Proposition 4.1 tells us ", "type": "text"}, {"bbox": [414, 440, 492, 452], "score": 0.94, "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [492, 435, 544, 459], "score": 1.0, "content": " for some", "type": "text"}], "index": 18}, {"bbox": [71, 454, 541, 469], "spans": [{"bbox": [71, 456, 89, 467], "score": 0.88, "content": "a,b", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [89, 454, 170, 469], "score": 1.0, "content": ", and we know ", "type": "text"}, {"bbox": [170, 455, 232, 466], "score": 0.93, "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [232, 454, 254, 469], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [255, 457, 293, 466], "score": 0.91, "content": "c=\\pm1", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [294, 454, 443, 469], "score": 1.0, "content": ". Hence from (2.7b) we get ", "type": "text"}, {"bbox": [443, 456, 489, 468], "score": 0.9, "content": "k b\\not\\equiv-1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [489, 454, 541, 469], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}, {"bbox": [71, 469, 542, 483], "spans": [{"bbox": [71, 469, 113, 483], "score": 1.0, "content": "Hitting ", "type": "text"}, {"bbox": [114, 474, 121, 480], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [121, 469, 153, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [154, 469, 210, 482], "score": 0.94, "content": "\\pi[-b]^{-1}C^{a}", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [211, 469, 340, 483], "score": 1.0, "content": ", we need consider only ", "type": "text"}, {"bbox": [340, 470, 394, 481], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [394, 469, 542, 483], "score": 1.0, "content": ". It is now immediate that", "type": "text"}], "index": 20}, {"bbox": [71, 484, 540, 498], "spans": [{"bbox": [71, 485, 122, 496], "score": 0.92, "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [122, 484, 206, 498], "score": 1.0, "content": ", by commuting ", "type": "text"}, {"bbox": [207, 488, 214, 494], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 484, 244, 498], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [244, 485, 254, 494], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [254, 484, 389, 498], "score": 1.0, "content": ". From (3.6a) we get that ", "type": "text"}, {"bbox": [389, 488, 397, 494], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [397, 484, 476, 498], "score": 1.0, "content": " must permute ", "type": "text"}, {"bbox": [476, 485, 490, 496], "score": 0.91, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [490, 484, 516, 498], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [516, 485, 536, 496], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [536, 484, 540, 498], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [71, 498, 541, 512], "spans": [{"bbox": [71, 498, 288, 512], "score": 1.0, "content": "Compare (3.6c) with (3.6d): since for any", "type": "text"}, {"bbox": [289, 499, 318, 510], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [318, 498, 541, 512], "score": 1.0, "content": " they have different numbers of summands,", "type": "text"}], "index": 22}, {"bbox": [71, 513, 539, 526], "spans": [{"bbox": [71, 513, 180, 526], "score": 1.0, "content": "we find in fact that ", "type": "text"}, {"bbox": [180, 517, 187, 523], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [188, 513, 262, 526], "score": 1.0, "content": " will fix both ", "type": "text"}, {"bbox": [262, 514, 276, 525], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [277, 513, 319, 526], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [320, 514, 334, 525], "score": 0.86, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [334, 513, 366, 526], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [367, 514, 387, 524], "score": 0.89, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [387, 513, 532, 526], "score": 1.0, "content": ". From (3.6b) we get that ", "type": "text"}, {"bbox": [532, 517, 539, 523], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}], "index": 23}, {"bbox": [70, 526, 541, 541], "spans": [{"bbox": [70, 526, 122, 541], "score": 1.0, "content": "permutes ", "type": "text"}, {"bbox": [122, 528, 136, 538], "score": 0.92, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [137, 526, 162, 541], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 528, 203, 539], "score": 0.93, "content": "\\Lambda_{1}+\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [203, 526, 345, 541], "score": 1.0, "content": ", and so (3.6d) now tells us ", "type": "text"}, {"bbox": [345, 528, 396, 539], "score": 0.92, "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [396, 526, 541, 541], "score": 1.0, "content": ". Finally, (3.6c) implies (for", "type": "text"}], "index": 24}, {"bbox": [71, 540, 540, 555], "spans": [{"bbox": [71, 542, 101, 553], "score": 0.88, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [101, 540, 108, 555], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [109, 542, 160, 553], "score": 0.92, "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [160, 540, 198, 555], "score": 1.0, "content": " (since ", "type": "text"}, {"bbox": [198, 542, 248, 551], "score": 0.88, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 9, "width": 50}, {"bbox": [249, 540, 367, 555], "score": 1.0, "content": "), and we are done for", "type": "text"}, {"bbox": [368, 542, 398, 553], "score": 0.9, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [398, 540, 437, 555], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [438, 542, 540, 554], "score": 0.93, "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "type": "inline_equation", "height": 12, "width": 102}], "index": 25}, {"bbox": [69, 555, 431, 569], "spans": [{"bbox": [69, 555, 199, 569], "score": 1.0, "content": "is a fusion-generator for ", "type": "text"}, {"bbox": [200, 557, 228, 565], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [229, 555, 255, 569], "score": 1.0, "content": " (see ", "type": "text"}, {"bbox": [256, 555, 280, 568], "score": 0.39, "content": "\\S2.2)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [280, 555, 397, 569], "score": 1.0, "content": ", we are also done for ", "type": "text"}, {"bbox": [397, 556, 426, 566], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [426, 555, 431, 569], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 22, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [69, 435, 544, 569]}, {"type": "text", "bbox": [70, 569, 542, 687], "lines": [{"bbox": [89, 564, 545, 589], "spans": [{"bbox": [89, 564, 117, 589], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 569, 139, 585], "score": 0.94, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [140, 564, 174, 589], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [175, 573, 205, 584], "score": 0.91, "content": "k\\geq7", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [205, 564, 313, 589], "score": 1.0, "content": ", (3.7a) tells us that ", "type": "text"}, {"bbox": [313, 571, 372, 585], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [372, 564, 508, 589], "score": 1.0, "content": " are permuted. For those ", "type": "text"}, {"bbox": [508, 573, 515, 582], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [516, 564, 545, 589], "score": 1.0, "content": ", the", "type": "text"}], "index": 27}, {"bbox": [71, 586, 541, 601], "spans": [{"bbox": [71, 586, 417, 601], "score": 1.0, "content": "highest multiplicities in (3.7b)\u2013(3.7d) are 3, 1, 2, respectively, so ", "type": "text"}, {"bbox": [417, 586, 475, 599], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [476, 586, 541, 601], "score": 1.0, "content": " must all be", "type": "text"}], "index": 28}, {"bbox": [70, 600, 541, 615], "spans": [{"bbox": [70, 600, 336, 615], "score": 1.0, "content": "fixed. The fusion product (3.7c) also tells us that ", "type": "text"}, {"bbox": [337, 600, 462, 613], "score": 0.92, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [462, 600, 541, 615], "score": 1.0, "content": " are permuted;", "type": "text"}], "index": 29}, {"bbox": [71, 614, 541, 630], "spans": [{"bbox": [71, 615, 227, 630], "score": 1.0, "content": "(3.7d) then says that the sets ", "type": "text"}, {"bbox": [227, 615, 272, 628], "score": 0.91, "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [273, 615, 279, 630], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [279, 615, 376, 628], "score": 0.91, "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [377, 615, 406, 630], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [406, 614, 541, 628], "score": 0.9, "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "type": "inline_equation", "height": 14, "width": 135}], "index": 30}, {"bbox": [71, 629, 541, 643], "spans": [{"bbox": [71, 630, 171, 642], "score": 0.9, "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [171, 630, 359, 643], "score": 1.0, "content": " are stabilised. Now (3.7b) implies ", "type": "text"}, {"bbox": [360, 629, 437, 642], "score": 0.91, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [437, 630, 541, 643], "score": 1.0, "content": " are all fixed, while", "type": "text"}], "index": 31}, {"bbox": [70, 643, 541, 659], "spans": [{"bbox": [70, 644, 111, 659], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [111, 644, 186, 657], "score": 0.92, "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [186, 644, 323, 659], "score": 1.0, "content": " is stabilised. Comparing ", "type": "text"}, {"bbox": [324, 643, 353, 657], "score": 0.25, "content": "(3.7\\mathrm{e})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [354, 644, 483, 659], "score": 1.0, "content": " and (3.7f), we get that ", "type": "text"}, {"bbox": [483, 643, 497, 656], "score": 0.87, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [498, 644, 541, 659], "score": 1.0, "content": " is fixed", "type": "text"}], "index": 32}, {"bbox": [66, 656, 540, 680], "spans": [{"bbox": [66, 656, 95, 680], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 662, 157, 673], "score": 0.92, "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [158, 656, 287, 680], "score": 1.0, "content": " are permuted. Finally, ", "type": "text"}, {"bbox": [288, 660, 318, 673], "score": 0.49, "content": "\\left(3.7\\mathrm{g}\\right)", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [319, 656, 357, 680], "score": 1.0, "content": " shows ", "type": "text"}, {"bbox": [357, 660, 371, 672], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [372, 656, 484, 680], "score": 1.0, "content": " also is fixed. To do ", "type": "text"}, {"bbox": [484, 658, 507, 674], "score": 0.93, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [508, 656, 540, 680], "score": 1.0, "content": "when", "type": "text"}], "index": 33}, {"bbox": [71, 674, 348, 691], "spans": [{"bbox": [71, 676, 100, 687], "score": 0.91, "content": "k\\leq6", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 674, 348, 691], "score": 1.0, "content": ", knowing q-dimensions really simplifies things.", "type": "text"}], "index": 34}], "index": 30.5, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [66, 564, 545, 691]}]}
0002044v1	22	8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. Anal. 6 (1996), 249–267. 9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteg-de Vries type equations, in: “Lie algebras and related topics”, Lecture Notes in Math, Vol. 933, Springer-Verlag, New York, 1982. 10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, “On axiomatic approaches to vertex operator algebras and modules”, Memoirs Amer. Math. Soc. 104 (1993). 11. J. Fr¨ohlich and T. Kerler, “Quantum groups, quantum categories and quantum field theory”, Lecture Notes in Mathematics, Vol. 1542, Springer-Verlag, Berlin, 1993. 12. J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991), 345–356. 13. J. Fuchs, B. Gato-Rivera, A. N. Schellekens and C. Schweigert, Modular invariants and fusion automorphisms from Galois theory, Phys. Lett. B334 (1994), 113–120. 14. J. Fuchs and P. van Driel, WZW fusion rules, quantum groups, and the modular matrix $$S$$ , Nucl. Phys. B346 (1990), 632–648. 15. J. Fuchs and P. van Driel, Fusion rule engineering, Lett. Math. Phys. 23 (1991), 11–18. 16. T. Gannon, WZW commutants, lattices, and level-one partition functions, Nucl. Phys. B396 (1993), 708–736; P. Ruelle, E. Thiran and J. Weyers, Implications of an arithmetical symmetry of the commutant for modular invariants, Nucl. Phys. B402 (1993), 693–708. 17. T. Gannon, Symmetries of the Kac-Peterson modular matrices of affine algebras, Invent. math. 122 (1995), 341–357. 18. T. Gannon, Ph. Ruelle and M. A. Walton, Automorphism modular invariants of current algebras, Commun. Math. Phys. 179 (1996), 121–156. 19. G. Georgiev and O. Mathieu, Cat´egorie de fusion pour les groupes de Chevalley, C. R. Acad. Sci. Paris 315 (1992), 659–662. 20. F. M. Goodman and H. Wenzl, Littlewood-Richardson coefficients for Hecke alge- bras at roots of unity, Adv. Math. 82 (1990), 244–265. 21. Y.-Z. Huang and J. Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113–134. 22. V. G. Kac, Simple Lie groups and the Legendre symbol, in: “Lie algebras, group theory, and partially ordered algebraic structures”, Lecture Notes in Math, Vol. 848, Springer-Verlag, Berlin, 1981. 23. V. G. Kac, “Infinite Dimensional Lie algebras”, 3rd edition, Cambridge University Press, Cambridge, 1990. 24. V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125–264. 25. V. G. Kac and M. Wakimoto, Modular and conformal constraints in representation theory of affine algebras, Adv. Math. 70 (1988), 156–236. 26. G. Lusztig, Exotic Fourier transform, Duke Math. J. 73 (1994), 227–241. 27. I. G. Macdonald, “Symmetric functions and Hall polynomials”, 2nd edition, Oxford University Press, New York, 1995.	<p>8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. 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Frenkel, Representations of affine Lie algebras, Hecke modular forms and\nKorteg-de Vries type equations, in: \u201cLie algebras and related topics\u201d, Lecture Notes\nin Math, Vol. 933, Springer-Verlag, New York, 1982.\n10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, \u201cOn axiomatic approaches to\nvertex operator algebras and modules\u201d, Memoirs Amer. Math. Soc. 104 (1993).\n11. J. Fr\u00a8ohlich and T. Kerler, \u201cQuantum groups, quantum categories and quantum\nfield theory\u201d, Lecture Notes in Mathematics, Vol. 1542, Springer-Verlag, Berlin, 1993.\n12. J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991), 345\u2013356.\n13. J. Fuchs, B. Gato-Rivera, A. N. Schellekens and C. Schweigert, Modular\ninvariants and fusion automorphisms from Galois theory, Phys. Lett. B334 (1994),\n113\u2013120.\n14. J. Fuchs and P. van Driel, WZW fusion rules, quantum groups, and the modular\nmatrix $$S$$ , Nucl. Phys. B346 (1990), 632\u2013648.\n15. J. 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0002044v1	13	At $$k=3$$ there is an order 3 Galois fusion-symmetry $$\pi_{3}=\pi\{5\}$$ , which sends $$J^{i}\Lambda_{1}\mapsto$$ $$J^{i}(2\Lambda_{6})\mapsto J^{i}\Lambda_{2}\mapsto J^{i}\Lambda_{1}$$ and fixes the other six weights. Theorem 3.E7. The only nontrivial fusion-symmetries for $${E}_{7}^{(1)}$$ are $$\pi[1]$$ at even $$k$$ , as well as $$\pi_{3}$$ and its inverse at $$k=3$$ . # 3.7. The algebra E8(1) A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+5\lambda_{4}+6\lambda_{5}+4\lambda_{6}+2\lambda_{7}+3\lambda_{8}$$ , and $$\kappa=k+30$$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $$k=2$$ , sending $$P_{+}=(0,\Lambda_{1},\Lambda_{7})$$ to $$(\Lambda_{7},\Lambda_{1},0)$$ , which plays no role in this paper (except in Theorem 5.1). The only fusion products we need can be derived from [28] and (2.4): $$(2\Lambda_{1})\vert\mathrm{\bf\sfXI}(2\Lambda_{1})=(0)_{4}$$ + $$(\Lambda_{1})_{5}$$ + $$(\Lambda_{2})_{5}$$ + $$(\Lambda_{3})_{4}$$ + $$(\Lambda_{7})_{4}$$ + $$2\,\mathbf{E}\left(2\Lambda_{1}\right)_{46}$$ + $$(2\Lambda_{2})_{6}$$ $$\Lambda_{1}$$ × $$\Lambda_{4}=(\Lambda_{3})_{5}$$ + $$(\Lambda_{4})_{6}$$ + $$(\Lambda_{5})_{6}$$ + $$(\Lambda_{6})_{5}$$ + $$(\Lambda_{1}+\Lambda_{3})_{6}$$ + $$(\Lambda_{1}+\Lambda_{4})_{7}$$ + $$(\Lambda_{1}+\Lambda_{6})_{6}$$ $$\Lambda_{1}$$ × $$(\Lambda_{1}\!+\!\Lambda_{3})=(\Lambda_{3})_{6}$$ + $$(\Lambda_{4})_{6}$$ + $$(\Lambda_{1}+\Lambda_{2})_{6}$$ + $$2\,\mathtt{H}\,(\Lambda_{1}+\Lambda_{3})_{67}$$ + $$(\Lambda_{1}+\Lambda_{4})_{7}$$ $$\Lambda_{1}$$ × $$(2\Lambda_{7})=\!(\Lambda_{6})_{4}$$ + $$(\Lambda_{1}+\Lambda_{7})_{4}$$ + $$(2\Lambda_{7})_{5}$$ + $$(\Lambda_{2}+\Lambda_{7})_{5}$$ + $$(\Lambda_{7}+\Lambda_{8})_{5}$$ + $$(\Lambda_{1}+2\Lambda_{7})_{6}$$ $$(3.7g)$$ A fusion-symmetry at $$k=4$$ , called $$\pi_{4}$$ , was first found in [15]. It interchanges $$\Lambda_{1}\leftrightarrow\Lambda_{6}$$ and fixes the other eight weights in $$P_{+}$$ . There also is a fusion-symmetry, called $$\pi_{5}$$ , at $$k=5$$ which interchanges $$\Lambda_{7}\leftrightarrow2\Lambda_{1}$$ , $$\Lambda_{8}\leftrightarrow\Lambda_{1}+\Lambda_{2}$$ , and $$\Lambda_{6}\leftrightarrow\Lambda_{2}+\Lambda_{7}$$ , and fixes the nine other weights. The exceptional $$\pi_{5}$$ is closely related to the Galois permutation $$\lambda\mapsto\lambda^{(13)}$$ . Theorem 3.E8. The only nontrivial fusion-symmetries for $${E}_{8}^{(1)}$$ are $$\pi_{4}$$ and $$\pi_{5}$$ , oc- curring at $$k=4$$ and 5 respectively. # 3.8. The algebra F 4(1) A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+2\lambda_{3}+\lambda_{4}$$ , and $$\kappa=k+9$$ . Again, the conjugations and simple-currents are trivial.	<p>At $$k=3$$ there is an order 3 Galois fusion-symmetry $$\pi_{3}=\pi\{5\}$$ , which sends $$J^{i}\Lambda_{1}\mapsto$$ $$J^{i}(2\Lambda_{6})\mapsto J^{i}\Lambda_{2}\mapsto J^{i}\Lambda_{1}$$ and fixes the other six weights.</p> <p>Theorem 3.E7. The only nontrivial fusion-symmetries for $${E}_{7}^{(1)}$$ are $$\pi[1]$$ at even $$k$$ , as well as $$\pi_{3}$$ and its inverse at $$k=3$$ .</p> <h1>3.7. The algebra E8(1)</h1> <p>A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+5\lambda_{4}+6\lambda_{5}+4\lambda_{6}+2\lambda_{7}+3\lambda_{8}$$ , and $$\kappa=k+30$$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $$k=2$$ , sending $$P_{+}=(0,\Lambda_{1},\Lambda_{7})$$ to $$(\Lambda_{7},\Lambda_{1},0)$$ , which plays no role in this paper (except in Theorem 5.1).</p> <p>The only fusion products we need can be derived from [28] and (2.4):</p> <p>$$(2\Lambda_{1})\vert\mathrm{\bf\sfXI}(2\Lambda_{1})=(0)_{4}$$ + $$(\Lambda_{1})_{5}$$ + $$(\Lambda_{2})_{5}$$ + $$(\Lambda_{3})_{4}$$ + $$(\Lambda_{7})_{4}$$ + $$2\,\mathbf{E}\left(2\Lambda_{1}\right)_{46}$$ + $$(2\Lambda_{2})_{6}$$</p> <p>$$\Lambda_{1}$$ × $$\Lambda_{4}=(\Lambda_{3})_{5}$$ + $$(\Lambda_{4})_{6}$$ + $$(\Lambda_{5})_{6}$$ + $$(\Lambda_{6})_{5}$$ + $$(\Lambda_{1}+\Lambda_{3})_{6}$$ + $$(\Lambda_{1}+\Lambda_{4})_{7}$$ + $$(\Lambda_{1}+\Lambda_{6})_{6}$$</p> <p>$$\Lambda_{1}$$ × $$(\Lambda_{1}\!+\!\Lambda_{3})=(\Lambda_{3})_{6}$$ + $$(\Lambda_{4})_{6}$$ + $$(\Lambda_{1}+\Lambda_{2})_{6}$$ + $$2\,\mathtt{H}\,(\Lambda_{1}+\Lambda_{3})_{67}$$ + $$(\Lambda_{1}+\Lambda_{4})_{7}$$</p> <p>$$\Lambda_{1}$$ × $$(2\Lambda_{7})=\!(\Lambda_{6})_{4}$$ + $$(\Lambda_{1}+\Lambda_{7})_{4}$$ + $$(2\Lambda_{7})_{5}$$ + $$(\Lambda_{2}+\Lambda_{7})_{5}$$ + $$(\Lambda_{7}+\Lambda_{8})_{5}$$ + $$(\Lambda_{1}+2\Lambda_{7})_{6}$$ $$(3.7g)$$</p> <p>A fusion-symmetry at $$k=4$$ , called $$\pi_{4}$$ , was first found in [15]. It interchanges $$\Lambda_{1}\leftrightarrow\Lambda_{6}$$ and fixes the other eight weights in $$P_{+}$$ . There also is a fusion-symmetry, called $$\pi_{5}$$ , at $$k=5$$ which interchanges $$\Lambda_{7}\leftrightarrow2\Lambda_{1}$$ , $$\Lambda_{8}\leftrightarrow\Lambda_{1}+\Lambda_{2}$$ , and $$\Lambda_{6}\leftrightarrow\Lambda_{2}+\Lambda_{7}$$ , and fixes the nine other weights. The exceptional $$\pi_{5}$$ is closely related to the Galois permutation $$\lambda\mapsto\lambda^{(13)}$$ .</p> <p>Theorem 3.E8. The only nontrivial fusion-symmetries for $${E}_{8}^{(1)}$$ are $$\pi_{4}$$ and $$\pi_{5}$$ , oc- curring at $$k=4$$ and 5 respectively.</p> <h1>3.8. The algebra F 4(1)</h1> <p>A weight $$\lambda$$ in $$P_{+}$$ satisfies $$k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+2\lambda_{3}+\lambda_{4}$$ , and $$\kappa=k+9$$ . Again, the conjugations and simple-currents are trivial.</p>	[{"type": "text", "coordinates": [71, 70, 541, 100], "content": "At $$k=3$$ there is an order 3 Galois fusion-symmetry $$\\pi_{3}=\\pi\\{5\\}$$ , which sends $$J^{i}\\Lambda_{1}\\mapsto$$\n$$J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}$$ and fixes the other six weights.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [71, 107, 541, 137], "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for $${E}_{7}^{(1)}$$ are $$\\pi[1]$$ at even $$k$$ ,\nas well as $$\\pi_{3}$$ and its inverse at $$k=3$$ .", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [70, 150, 183, 167], "content": "3.7. The algebra E8(1)", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [70, 173, 542, 230], "content": "A weight $$\\lambda$$ in $$P_{+}$$ satisfies $$k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}$$ , and\n$$\\kappa=k+30$$ . The conjugations and simple-currents are all trivial, except for an anomolous\nsimple-current at $$k=2$$ , sending $$P_{+}=(0,\\Lambda_{1},\\Lambda_{7})$$ to $$(\\Lambda_{7},\\Lambda_{1},0)$$ , which plays no role in this\npaper (except in Theorem 5.1).", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [94, 230, 460, 245], "content": "The only fusion products we need can be derived from [28] and (2.4):", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [47, 364, 469, 381], "content": "$$(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}$$ + $$(\\Lambda_{1})_{5}$$ + $$(\\Lambda_{2})_{5}$$ + $$(\\Lambda_{3})_{4}$$ + $$(\\Lambda_{7})_{4}$$ + $$2\\,\\mathbf{E}\\left(2\\Lambda_{1}\\right)_{46}$$ + $$(2\\Lambda_{2})_{6}$$", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [73, 418, 520, 435], "content": "$$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{4}=(\\Lambda_{3})_{5}$$ + $$(\\Lambda_{4})_{6}$$ + $$(\\Lambda_{5})_{6}$$ + $$(\\Lambda_{6})_{5}$$ + $$(\\Lambda_{1}+\\Lambda_{3})_{6}$$ + $$(\\Lambda_{1}+\\Lambda_{4})_{7}$$ + $$(\\Lambda_{1}+\\Lambda_{6})_{6}$$", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [45, 454, 460, 470], "content": "$$\\Lambda_{1}$$ \u00d7 $$(\\Lambda_{1}\\!+\\!\\Lambda_{3})=(\\Lambda_{3})_{6}$$ + $$(\\Lambda_{4})_{6}$$ + $$(\\Lambda_{1}+\\Lambda_{2})_{6}$$ + $$2\\,\\mathtt{H}\\,(\\Lambda_{1}+\\Lambda_{3})_{67}$$ + $$(\\Lambda_{1}+\\Lambda_{4})_{7}$$", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [59, 526, 530, 543], "content": "$$\\Lambda_{1}$$ \u00d7 $$(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}$$ + $$(\\Lambda_{1}+\\Lambda_{7})_{4}$$ + $$(2\\Lambda_{7})_{5}$$ + $$(\\Lambda_{2}+\\Lambda_{7})_{5}$$ + $$(\\Lambda_{7}+\\Lambda_{8})_{5}$$ + $$(\\Lambda_{1}+2\\Lambda_{7})_{6}$$ $$(3.7g)$$", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 554, 542, 613], "content": "A fusion-symmetry at $$k=4$$ , called $$\\pi_{4}$$ , was first found in [15]. It interchanges $$\\Lambda_{1}\\leftrightarrow\\Lambda_{6}$$\nand fixes the other eight weights in $$P_{+}$$ . There also is a fusion-symmetry, called $$\\pi_{5}$$ , at $$k=5$$\nwhich interchanges $$\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}$$ , $$\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}$$ , and $$\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}$$ , and fixes the nine other\nweights. The exceptional $$\\pi_{5}$$ is closely related to the Galois permutation $$\\lambda\\mapsto\\lambda^{(13)}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [70, 618, 541, 650], "content": "Theorem 3.E8. The only nontrivial fusion-symmetries for $${E}_{8}^{(1)}$$ are $$\\pi_{4}$$ and $$\\pi_{5}$$ , oc-\ncurring at $$k=4$$ and 5 respectively.", "block_type": "text", "index": 11}, {"type": "title", "coordinates": [71, 663, 183, 680], "content": "3.8. The algebra F 4(1)", "block_type": "title", "index": 12}, {"type": "text", "coordinates": [70, 686, 541, 715], "content": "A weight $$\\lambda$$ in $$P_{+}$$ satisfies $$k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}$$ , and $$\\kappa=k+9$$ . Again, the\nconjugations and simple-currents are trivial.", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [94, 73, 111, 90], "content": "At", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [112, 75, 141, 84], "content": "k=3", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [141, 73, 371, 90], "content": " there is an order 3 Galois fusion-symmetry ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [372, 73, 426, 87], "content": "\\pi_{3}=\\pi\\{5\\}", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [426, 73, 498, 90], "content": ", which sends ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [498, 73, 541, 86], "content": "J^{i}\\Lambda_{1}\\mapsto", "score": 0.89, "index": 6}, {"type": "inline_equation", "coordinates": [71, 88, 200, 101], "content": "J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [200, 87, 367, 103], "content": " and fixes the other six weights.", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [92, 109, 414, 128], "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [414, 108, 438, 126], "content": "{E}_{7}^{(1)}", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [438, 109, 461, 128], "content": "are ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [461, 110, 483, 126], "content": "\\pi[1]", "score": 0.71, "index": 12}, {"type": "text", "coordinates": [483, 109, 528, 128], "content": " at even ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [529, 112, 536, 123], "content": "k", "score": 0.73, "index": 14}, {"type": "text", "coordinates": [536, 109, 540, 128], "content": ",", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [72, 126, 126, 139], "content": "as well as ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [126, 129, 139, 138], "content": "\\pi_{3}", "score": 0.75, "index": 17}, {"type": "text", "coordinates": [139, 126, 237, 139], "content": " and its inverse at ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [238, 128, 267, 137], "content": "k=3", "score": 0.91, "index": 19}, {"type": "text", "coordinates": [268, 126, 271, 139], "content": ".", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [67, 148, 189, 174], "content": "3.7. The algebra E8(1)", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [95, 176, 144, 190], "content": "A weight ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [144, 177, 153, 186], "content": "\\lambda", "score": 0.81, "index": 23}, {"type": "text", "coordinates": [153, 176, 168, 190], "content": " in ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [168, 176, 185, 189], "content": "P_{+}", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [185, 176, 231, 190], "content": " satisfies ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [231, 176, 514, 188], "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}", "score": 0.85, "index": 27}, {"type": "text", "coordinates": [514, 176, 541, 190], "content": ", and", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [71, 190, 128, 202], "content": "\\kappa=k+30", "score": 0.89, "index": 29}, {"type": "text", "coordinates": [128, 190, 541, 205], "content": ". 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232], "content": "paper (except in Theorem 5.1).", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [95, 232, 458, 247], "content": "The only fusion products we need can be derived from [28] and (2.4):", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [48, 366, 158, 382], "content": "(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}", "score": 0.73, "index": 40}, {"type": "text", "coordinates": [158, 368, 174, 385], "content": " + ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [174, 366, 204, 381], "content": "(\\Lambda_{1})_{5}", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [204, 368, 221, 385], "content": " + ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [221, 366, 250, 381], "content": "(\\Lambda_{2})_{5}", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [251, 368, 267, 385], "content": " + ", "score": 1.0, "index": 45}, {"type": "inline_equation", 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702], "content": "P_{+}", "caption": ""}, {"type": "inline", "coordinates": [234, 689, 396, 701], "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}", "caption": ""}, {"type": "inline", "coordinates": [425, 690, 476, 700], "content": "\\kappa=k+9", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "At $k=3$ there is an order 3 Galois fusion-symmetry $\\pi_{3}=\\pi\\{5\\}$ , which sends $J^{i}\\Lambda_{1}\\mapsto$ $J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}$ and fixes the other six weights. ", "page_idx": 13}, {"type": "text", "text": "Theorem 3.E7. The only nontrivial fusion-symmetries for ${E}_{7}^{(1)}$ are $\\pi[1]$ at even $k$ , as well as $\\pi_{3}$ and its inverse at $k=3$ . ", "page_idx": 13}, {"type": "text", "text": "3.7. The algebra E8(1) ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}$ , and $\\kappa=k+30$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $k=2$ , sending $P_{+}=(0,\\Lambda_{1},\\Lambda_{7})$ to $(\\Lambda_{7},\\Lambda_{1},0)$ , which plays no role in this paper (except in Theorem 5.1). ", "page_idx": 13}, {"type": "text", "text": "The only fusion products we need can be derived from [28] and (2.4): ", "page_idx": 13}, {"type": "text", "text": "$(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}$ + $(\\Lambda_{1})_{5}$ + $(\\Lambda_{2})_{5}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{7})_{4}$ + $2\\,\\mathbf{E}\\left(2\\Lambda_{1}\\right)_{46}$ + $(2\\Lambda_{2})_{6}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{5}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{5})_{6}$ + $(\\Lambda_{6})_{5}$ + $(\\Lambda_{1}+\\Lambda_{3})_{6}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ + $(\\Lambda_{1}+\\Lambda_{6})_{6}$ $\\Lambda_{1}$ \u00d7 $(\\Lambda_{1}\\!+\\!\\Lambda_{3})=(\\Lambda_{3})_{6}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{1}+\\Lambda_{2})_{6}$ + $2\\,\\mathtt{H}\\,(\\Lambda_{1}+\\Lambda_{3})_{67}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ $\\Lambda_{1}$ \u00d7 $(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}$ + $(\\Lambda_{1}+\\Lambda_{7})_{4}$ + $(2\\Lambda_{7})_{5}$ + $(\\Lambda_{2}+\\Lambda_{7})_{5}$ + $(\\Lambda_{7}+\\Lambda_{8})_{5}$ + $(\\Lambda_{1}+2\\Lambda_{7})_{6}$ $(3.7g)$ ", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "A fusion-symmetry at $k=4$ , called $\\pi_{4}$ , was first found in [15]. It interchanges $\\Lambda_{1}\\leftrightarrow\\Lambda_{6}$ and fixes the other eight weights in $P_{+}$ . There also is a fusion-symmetry, called $\\pi_{5}$ , at $k=5$ which interchanges $\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}$ , $\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}$ , and $\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}$ , and fixes the nine other weights. The exceptional $\\pi_{5}$ is closely related to the Galois permutation $\\lambda\\mapsto\\lambda^{(13)}$ . ", "page_idx": 13}, {"type": "text", "text": "Theorem 3.E8. The only nontrivial fusion-symmetries for ${E}_{8}^{(1)}$ are $\\pi_{4}$ and $\\pi_{5}$ , occurring at $k=4$ and 5 respectively. ", "page_idx": 13}, {"type": "text", "text": "3.8. The algebra F 4(1) ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}$ , and $\\kappa=k+9$ . Again, the conjugations and simple-currents are trivial. 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It interchanges ", "type": "text"}, {"bbox": [493, 556, 540, 569], "score": 0.92, "content": "\\Lambda_{1}\\leftrightarrow\\Lambda_{6}", "type": "inline_equation", "height": 13, "width": 47}], "index": 14}, {"bbox": [71, 571, 540, 586], "spans": [{"bbox": [71, 571, 252, 586], "score": 1.0, "content": "and fixes the other eight weights in ", "type": "text"}, {"bbox": [252, 571, 268, 585], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [269, 571, 477, 586], "score": 1.0, "content": ". There also is a fusion-symmetry, called", "type": "text"}, {"bbox": [478, 572, 491, 584], "score": 0.85, "content": "\\pi_{5}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [491, 571, 510, 586], "score": 1.0, "content": ", at ", "type": "text"}, {"bbox": [510, 571, 540, 583], "score": 0.88, "content": "k=5", "type": "inline_equation", "height": 12, "width": 30}], "index": 15}, {"bbox": [72, 585, 541, 600], "spans": [{"bbox": [72, 586, 173, 600], "score": 1.0, "content": "which interchanges ", "type": "text"}, {"bbox": [173, 585, 226, 598], "score": 0.93, "content": "\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [226, 586, 232, 600], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 585, 306, 598], "score": 0.91, "content": "\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 586, 335, 600], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [335, 585, 410, 598], "score": 0.93, "content": "\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [410, 586, 541, 600], "score": 1.0, "content": ", and fixes the nine other", "type": "text"}], "index": 16}, {"bbox": [69, 598, 508, 615], "spans": [{"bbox": [69, 599, 205, 615], "score": 1.0, "content": "weights. The exceptional ", "type": "text"}, {"bbox": [205, 602, 218, 613], "score": 0.88, "content": "\\pi_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [219, 599, 452, 615], "score": 1.0, "content": " is closely related to the Galois permutation ", "type": "text"}, {"bbox": [452, 598, 503, 611], "score": 0.91, "content": "\\lambda\\mapsto\\lambda^{(13)}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [503, 599, 508, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15.5}, {"type": "text", "bbox": [70, 618, 541, 650], "lines": [{"bbox": [90, 619, 540, 641], "spans": [{"bbox": [90, 619, 415, 641], "score": 1.0, "content": "Theorem 3.E8. 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Again, the", "type": "text"}], "index": 21}, {"bbox": [72, 703, 303, 717], "spans": [{"bbox": [72, 703, 303, 717], "score": 1.0, "content": "conjugations and simple-currents are trivial.", "type": "text"}], "index": 22}], "index": 21.5}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 313, 744], "spans": [{"bbox": [298, 731, 313, 744], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 541, 100], "lines": [{"bbox": [94, 73, 541, 90], "spans": [{"bbox": [94, 73, 111, 90], "score": 1.0, "content": "At", "type": "text"}, {"bbox": [112, 75, 141, 84], "score": 0.9, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [141, 73, 371, 90], "score": 1.0, "content": " there is an order 3 Galois fusion-symmetry ", "type": "text"}, {"bbox": [372, 73, 426, 87], "score": 0.93, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 54}, {"bbox": [426, 73, 498, 90], "score": 1.0, "content": ", which sends ", "type": "text"}, {"bbox": [498, 73, 541, 86], "score": 0.89, "content": "J^{i}\\Lambda_{1}\\mapsto", "type": "inline_equation", "height": 13, "width": 43}], "index": 0}, {"bbox": [71, 87, 367, 103], "spans": [{"bbox": [71, 88, 200, 101], "score": 0.92, "content": "J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [200, 87, 367, 103], "score": 1.0, "content": " and fixes the other six weights.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [71, 73, 541, 103]}, {"type": "text", "bbox": [71, 107, 541, 137], "lines": [{"bbox": [92, 108, 540, 128], "spans": [{"bbox": [92, 109, 414, 128], "score": 1.0, "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for ", "type": "text"}, {"bbox": [414, 108, 438, 126], "score": 0.9, "content": "{E}_{7}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [438, 109, 461, 128], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [461, 110, 483, 126], "score": 0.71, "content": "\\pi[1]", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 109, 528, 128], "score": 1.0, "content": " at even ", "type": "text"}, {"bbox": [529, 112, 536, 123], "score": 0.73, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [536, 109, 540, 128], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [72, 126, 271, 139], "spans": [{"bbox": [72, 126, 126, 139], "score": 1.0, "content": "as well as ", "type": "text"}, {"bbox": [126, 129, 139, 138], "score": 0.75, "content": "\\pi_{3}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 126, 237, 139], "score": 1.0, "content": " and its inverse at ", "type": "text"}, {"bbox": [238, 128, 267, 137], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [268, 126, 271, 139], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 108, 540, 139]}, {"type": "title", "bbox": [70, 150, 183, 167], "lines": [{"bbox": [67, 148, 189, 174], "spans": [{"bbox": [67, 148, 189, 174], "score": 1.0, "content": "3.7. The algebra E8(1)", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 173, 542, 230], "lines": [{"bbox": [95, 176, 541, 190], "spans": [{"bbox": [95, 176, 144, 190], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 177, 153, 186], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [153, 176, 168, 190], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [168, 176, 185, 189], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [185, 176, 231, 190], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [231, 176, 514, 188], "score": 0.85, "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}", "type": "inline_equation", "height": 12, "width": 283}, {"bbox": [514, 176, 541, 190], "score": 1.0, "content": ", and", "type": "text"}], "index": 5}, {"bbox": [71, 190, 541, 205], "spans": [{"bbox": [71, 190, 128, 202], "score": 0.89, "content": "\\kappa=k+30", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [128, 190, 541, 205], "score": 1.0, "content": ". 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It interchanges ", "type": "text"}, {"bbox": [493, 556, 540, 569], "score": 0.92, "content": "\\Lambda_{1}\\leftrightarrow\\Lambda_{6}", "type": "inline_equation", "height": 13, "width": 47}], "index": 14}, {"bbox": [71, 571, 540, 586], "spans": [{"bbox": [71, 571, 252, 586], "score": 1.0, "content": "and fixes the other eight weights in ", "type": "text"}, {"bbox": [252, 571, 268, 585], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [269, 571, 477, 586], "score": 1.0, "content": ". There also is a fusion-symmetry, called", "type": "text"}, {"bbox": [478, 572, 491, 584], "score": 0.85, "content": "\\pi_{5}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [491, 571, 510, 586], "score": 1.0, "content": ", at ", "type": "text"}, {"bbox": [510, 571, 540, 583], "score": 0.88, "content": "k=5", "type": "inline_equation", "height": 12, "width": 30}], "index": 15}, {"bbox": [72, 585, 541, 600], "spans": [{"bbox": [72, 586, 173, 600], "score": 1.0, "content": "which interchanges ", "type": "text"}, {"bbox": [173, 585, 226, 598], "score": 0.93, "content": "\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [226, 586, 232, 600], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 585, 306, 598], "score": 0.91, "content": "\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 586, 335, 600], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [335, 585, 410, 598], "score": 0.93, "content": "\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [410, 586, 541, 600], "score": 1.0, "content": ", and fixes the nine other", "type": "text"}], "index": 16}, {"bbox": [69, 598, 508, 615], "spans": [{"bbox": [69, 599, 205, 615], "score": 1.0, "content": "weights. The exceptional ", "type": "text"}, {"bbox": [205, 602, 218, 613], "score": 0.88, "content": "\\pi_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [219, 599, 452, 615], "score": 1.0, "content": " is closely related to the Galois permutation ", "type": "text"}, {"bbox": [452, 598, 503, 611], "score": 0.91, "content": "\\lambda\\mapsto\\lambda^{(13)}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [503, 599, 508, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [69, 556, 541, 615]}, {"type": "text", "bbox": [70, 618, 541, 650], "lines": [{"bbox": [90, 619, 540, 641], "spans": [{"bbox": [90, 619, 415, 641], "score": 1.0, "content": "Theorem 3.E8. The only nontrivial fusion-symmetries for ", "type": "text"}, {"bbox": [415, 619, 438, 636], "score": 0.91, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [438, 619, 462, 641], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [462, 624, 475, 635], "score": 0.78, "content": "\\pi_{4}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [476, 619, 502, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [503, 624, 516, 635], "score": 0.79, "content": "\\pi_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [516, 619, 540, 641], "score": 1.0, "content": ", oc-", "type": "text"}], "index": 18}, {"bbox": [72, 637, 257, 653], "spans": [{"bbox": [72, 637, 127, 653], "score": 1.0, "content": "curring at ", "type": "text"}, {"bbox": [128, 639, 157, 648], "score": 0.85, "content": "k=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [157, 637, 257, 653], "score": 1.0, "content": " and 5 respectively.", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 619, 540, 653]}, {"type": "title", "bbox": [71, 663, 183, 680], "lines": [{"bbox": [69, 663, 186, 683], "spans": [{"bbox": [69, 663, 186, 683], "score": 1.0, "content": "3.8. The algebra F 4(1)", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 686, 541, 715], "lines": [{"bbox": [94, 688, 541, 703], "spans": [{"bbox": [94, 688, 145, 703], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 690, 153, 699], "score": 0.86, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [154, 688, 171, 703], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [171, 690, 186, 702], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [187, 688, 234, 703], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [234, 689, 396, 701], "score": 0.93, "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}", "type": "inline_equation", "height": 12, "width": 162}, {"bbox": [396, 688, 425, 703], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [425, 690, 476, 700], "score": 0.89, "content": "\\kappa=k+9", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [477, 688, 541, 703], "score": 1.0, "content": ". Again, the", "type": "text"}], "index": 21}, {"bbox": [72, 703, 303, 717], "spans": [{"bbox": [72, 703, 303, 717], "score": 1.0, "content": "conjugations and simple-currents are trivial.", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 688, 541, 717]}]}
0002044v1	18	$$a=a^{\prime}=0$$ holds for $$k r$$ even. From the fusion $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ we get $$\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$$ if $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ ; for $$r<k$$ conclude the argument with the calculation as in $$\S4.3$$ . When $$r=k$$ , that inequality only holds for $$\ell>1$$ , but we can force $$\pi\Lambda_{2}=\Lambda_{2}$$ by hitting $$\pi$$ if necessary with $$\pi_{\mathrm{rld}}$$ . The remaining case $$C_{2,3}$$ follows because $$\pi^{\prime}J0\,=\,J0$$ : by (2.7b) $$\pi\Lambda_{1}\notin S\Lambda_{2}$$ , and by (2.7a) $$\pi\Lambda_{1}\neq3\Lambda_{1}$$ ( $$3\Lambda_{1}$$ is a $$J$$ -fixed-point). # 4.5. The $$D$$ -series argument $$k=1$$ is trivial, and $$k=2$$ will be considered shortly. For $$k>2$$ , Proposition 4.1 tells us that $$\pi\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ and $$\pi^{\prime}\Lambda_{1}=J_{v}^{a^{\prime}}J_{s}^{b^{\prime}}\Lambda_{1}$$ , for $$a,a^{\prime},b,b^{\prime}\in\{0,1\}$$ . Immediate from (3.4) is that $$\chi_{\Lambda_{1}}[\Lambda_{1}]>0$$ and that $$\chi_{\Lambda_{1}}[\psi]$$ , for a spinor $$\psi$$ , takes its maximum at $$C^{i}J_{v}^{j}\Lambda_{r}$$ . Our first step is to force $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ . Unfortunately this requires a case analysis. Consider first even $$r\,\neq\,4$$ , and even $$k\ >\ 2$$ . Now, $$0\;\neq\;S_{\Lambda_{1}\Lambda_{1}}\;=\;S_{\pi\Lambda_{1},\pi^{\prime}\Lambda_{1}}$$ forces $$b=b^{\prime}$$ ; hence hitting with the simple-current automorphism $$\pi\left[{\begin{array}{l l}{0}&{a}\\ {a^{\prime}}&{b}\end{array}}\right]$$ , we may assume $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ . Next consider even $$r\neq4$$ and odd $$k\,>\,2$$ . Either of $$\pi\Lambda_{1}=J_{v}\Lambda_{1}$$ or $$\pi^{\prime}\Lambda_{1}\,=\,J_{v}\Lambda_{1}$$ is impossible, by comparing $$S_{\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\Lambda_{1},J0}$$ for any simple-current $$J$$ . For any of the three remaining choices of $$J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ , we can find a simple-current automorphism of the form $$\pi\left[{\ast}\quad a\,\right];$$ hitting its inverse onto $$\pi$$ allows us to take $$a=b=0$$ . Again $$0\not=S_{\Lambda_{1}\Lambda_{1}}$$ forces $$b^{\prime}=0$$ , and now $$a^{\prime}=1$$ is forbidden. Thus again $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ . As usual, $$r=4$$ is complicated by triality. We can force $$\pi\Lambda_{1}=\Lambda_{1}$$ . That we can also take $$\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ , follows from the inequality $$\chi_{\Lambda_{1}}[\Lambda_{1}]>\chi_{\Lambda_{1}}[\Lambda_{3}]=\chi_{\Lambda_{1}}[\Lambda_{4}]>0$$ , valid for $$k\geq3$$ . Establishing that inequality from (3.4) is equivalent to showing $$1+\cos(x)+\cos(2x)+\cos(4x)>\cos(x/2)+\cos(3x/2)+\cos(5x/2)+\cos(7x/2)>0$$ 2) for $$0<x\le2\pi/9$$ , which can be shown e.g. using Taylor series. For odd $$r$$ , the charge-conjugation $$C$$ equals $$C_{1}$$ . Since it must commute with $$\pi$$ , i.e. that $$C_{1}\pi\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\Lambda_{1}$$ must equal $$\pi C_{1}\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ , we get that $$b=0$$ . Similarly $$b^{\prime}=0$$ . When $$k$$ is odd, eliminate $$a=1$$ and $$a\prime=1$$ by comparing $$S_{\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\Lambda_{1},J0}$$ as before. The hardest case is $$k$$ even. We can force $$\pi\Lambda_{1}\,=\,\Lambda_{1}$$ by hitting with $$\pi[a]$$ . Suppose for contradiction that $$\pi^{\prime}\Lambda_{1}=J_{v}\Lambda_{1}$$ . We know $$\pi^{\prime}(J_{v}0)=J_{v}0$$ (compare $$S_{\Lambda_{1},J_{v}0}$$ and $$S_{\Lambda_{1},J0})$$ , so by (2.7b) $$\pi\Lambda_{r}$$ must be a spinor. $$\chi_{\Lambda_{1}}[\Lambda_{r}]\;=\;\chi_{J_{v}\Lambda_{1}}[\pi\Lambda_{r}]$$ requires $$\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r}$$ . From the $$\Lambda_{1}\boxtimes\Lambda_{r}$$ fusion we get $$\pi\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r-1}$$ , but $$C\pi=\pi C$$ says that $$\pi\Lambda_{r-1}=$$ $$C_{1}^{i}J_{v}^{j+1}J_{s}\Lambda_{r-1}$$ — a contradiction. Thus in all cases we have $$\pi\Lambda_{1}\,=\,\pi^{\prime}\Lambda_{1}\,=\,\Lambda_{1}$$ . We know $$\pi^{\prime}(J_{v}0)\;=\;J_{v}0$$ (compare $$S_{\Lambda_{1},J_{v}0}$$ and $$S_{\Lambda_{1},J0})$$ , so $$\pi\Lambda_{r}$$ is a spinor and in fact must equal $$\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}\Lambda_{r}$$ . Hitting with $$(C_{1}^{i}\pi_{v}^{j})^{-1}$$ , we can require $$\pi\Lambda_{r}=\Lambda_{r}$$ . That $$\pi\Lambda_{r-1}$$ must now equal $$\Lambda_{r-1}$$ follows from the $$\Lambda_{1}\boxtimes\Lambda_{r}$$ fusion.	<p>$$a=a^{\prime}=0$$ holds for $$k r$$ even. From the fusion $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ we get $$\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$$ if $$\pi\Lambda_{\ell}=\Lambda_{\ell}$$ ; for $$r<k$$ conclude the argument with the calculation</p> <p>as in $$\S4.3$$ . When $$r=k$$ , that inequality only holds for $$\ell>1$$ , but we can force $$\pi\Lambda_{2}=\Lambda_{2}$$ by hitting $$\pi$$ if necessary with $$\pi_{\mathrm{rld}}$$ .</p> <p>The remaining case $$C_{2,3}$$ follows because $$\pi^{\prime}J0\,=\,J0$$ : by (2.7b) $$\pi\Lambda_{1}\notin S\Lambda_{2}$$ , and by (2.7a) $$\pi\Lambda_{1}\neq3\Lambda_{1}$$ ( $$3\Lambda_{1}$$ is a $$J$$ -fixed-point).</p> <h1>4.5. The $$D$$ -series argument</h1> <p>$$k=1$$ is trivial, and $$k=2$$ will be considered shortly. For $$k>2$$ , Proposition 4.1 tells us that $$\pi\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ and $$\pi^{\prime}\Lambda_{1}=J_{v}^{a^{\prime}}J_{s}^{b^{\prime}}\Lambda_{1}$$ , for $$a,a^{\prime},b,b^{\prime}\in\{0,1\}$$ . Immediate from (3.4) is that $$\chi_{\Lambda_{1}}[\Lambda_{1}]>0$$ and that $$\chi_{\Lambda_{1}}[\psi]$$ , for a spinor $$\psi$$ , takes its maximum at $$C^{i}J_{v}^{j}\Lambda_{r}$$ . Our first step is to force $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ . Unfortunately this requires a case analysis.</p> <p>Consider first even $$r\,\neq\,4$$ , and even $$k\ >\ 2$$ . Now, $$0\;\neq\;S_{\Lambda_{1}\Lambda_{1}}\;=\;S_{\pi\Lambda_{1},\pi^{\prime}\Lambda_{1}}$$ forces $$b=b^{\prime}$$ ; hence hitting with the simple-current automorphism $$\pi\left[{\begin{array}{l l}{0}&{a}\\ {a^{\prime}}&{b}\end{array}}\right]$$ , we may assume $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ .</p> <p>Next consider even $$r\neq4$$ and odd $$k\,>\,2$$ . Either of $$\pi\Lambda_{1}=J_{v}\Lambda_{1}$$ or $$\pi^{\prime}\Lambda_{1}\,=\,J_{v}\Lambda_{1}$$ is impossible, by comparing $$S_{\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\Lambda_{1},J0}$$ for any simple-current $$J$$ . For any of the three remaining choices of $$J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ , we can find a simple-current automorphism of the form $$\pi\left[{\ast}\quad a\,\right];$$ hitting its inverse onto $$\pi$$ allows us to take $$a=b=0$$ . Again $$0\not=S_{\Lambda_{1}\Lambda_{1}}$$ forces $$b^{\prime}=0$$ , and now $$a^{\prime}=1$$ is forbidden. Thus again $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ .</p> <p>As usual, $$r=4$$ is complicated by triality. We can force $$\pi\Lambda_{1}=\Lambda_{1}$$ . That we can also take $$\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ , follows from the inequality $$\chi_{\Lambda_{1}}[\Lambda_{1}]>\chi_{\Lambda_{1}}[\Lambda_{3}]=\chi_{\Lambda_{1}}[\Lambda_{4}]>0$$ , valid for $$k\geq3$$ . Establishing that inequality from (3.4) is equivalent to showing</p> <p>$$1+\cos(x)+\cos(2x)+\cos(4x)>\cos(x/2)+\cos(3x/2)+\cos(5x/2)+\cos(7x/2)>0$$ 2)</p> <p>for $$0<x\le2\pi/9$$ , which can be shown e.g. using Taylor series.</p> <p>For odd $$r$$ , the charge-conjugation $$C$$ equals $$C_{1}$$ . Since it must commute with $$\pi$$ , i.e. that $$C_{1}\pi\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\Lambda_{1}$$ must equal $$\pi C_{1}\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$$ , we get that $$b=0$$ . Similarly $$b^{\prime}=0$$ . When $$k$$ is odd, eliminate $$a=1$$ and $$a\prime=1$$ by comparing $$S_{\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\Lambda_{1},J0}$$ as before. The hardest case is $$k$$ even. We can force $$\pi\Lambda_{1}\,=\,\Lambda_{1}$$ by hitting with $$\pi[a]$$ . Suppose for contradiction that $$\pi^{\prime}\Lambda_{1}=J_{v}\Lambda_{1}$$ . We know $$\pi^{\prime}(J_{v}0)=J_{v}0$$ (compare $$S_{\Lambda_{1},J_{v}0}$$ and $$S_{\Lambda_{1},J0})$$ , so by (2.7b) $$\pi\Lambda_{r}$$ must be a spinor. $$\chi_{\Lambda_{1}}[\Lambda_{r}]\;=\;\chi_{J_{v}\Lambda_{1}}[\pi\Lambda_{r}]$$ requires $$\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r}$$ . From the $$\Lambda_{1}\boxtimes\Lambda_{r}$$ fusion we get $$\pi\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r-1}$$ , but $$C\pi=\pi C$$ says that $$\pi\Lambda_{r-1}=$$ $$C_{1}^{i}J_{v}^{j+1}J_{s}\Lambda_{r-1}$$ — a contradiction.</p> <p>Thus in all cases we have $$\pi\Lambda_{1}\,=\,\pi^{\prime}\Lambda_{1}\,=\,\Lambda_{1}$$ . We know $$\pi^{\prime}(J_{v}0)\;=\;J_{v}0$$ (compare $$S_{\Lambda_{1},J_{v}0}$$ and $$S_{\Lambda_{1},J0})$$ , so $$\pi\Lambda_{r}$$ is a spinor and in fact must equal $$\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}\Lambda_{r}$$ . Hitting with $$(C_{1}^{i}\pi_{v}^{j})^{-1}$$ , we can require $$\pi\Lambda_{r}=\Lambda_{r}$$ . That $$\pi\Lambda_{r-1}$$ must now equal $$\Lambda_{r-1}$$ follows from the $$\Lambda_{1}\boxtimes\Lambda_{r}$$ fusion.</p>	[{"type": "text", "coordinates": [69, 70, 542, 100], "content": "$$a=a^{\\prime}=0$$ holds for $$k r$$ even. From the fusion $$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{\\ell}$$ we get $$\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}$$ if\n$$\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$$ ; for $$r<k$$ conclude the argument with the calculation", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [99, 113, 500, 141], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 149, 541, 178], "content": "as in $$\\S4.3$$ . When $$r=k$$ , that inequality only holds for $$\\ell>1$$ , but we can force $$\\pi\\Lambda_{2}=\\Lambda_{2}$$\nby hitting $$\\pi$$ if necessary with $$\\pi_{\\mathrm{rld}}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 179, 541, 208], "content": "The remaining case $$C_{2,3}$$ follows because $$\\pi^{\\prime}J0\\,=\\,J0$$ : by (2.7b) $$\\pi\\Lambda_{1}\\notin S\\Lambda_{2}$$ , and by\n(2.7a) $$\\pi\\Lambda_{1}\\neq3\\Lambda_{1}$$ ( $$3\\Lambda_{1}$$ is a $$J$$ -fixed-point).", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [71, 221, 219, 236], "content": "4.5. The $$D$$ -series argument", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [70, 243, 541, 300], "content": "$$k=1$$ is trivial, and $$k=2$$ will be considered shortly. For $$k>2$$ , Proposition 4.1 tells\nus that $$\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$$ and $$\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}$$ , for $$a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}$$ . Immediate from (3.4)\nis that $$\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$$ and that $$\\chi_{\\Lambda_{1}}[\\psi]$$ , for a spinor $$\\psi$$ , takes its maximum at $$C^{i}J_{v}^{j}\\Lambda_{r}$$ . Our\nfirst step is to force $$\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$$ . Unfortunately this requires a case analysis.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [69, 301, 541, 358], "content": "Consider first even $$r\\,\\neq\\,4$$ , and even $$k\\ >\\ 2$$ . Now, $$0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}$$ forces\n$$b=b^{\\prime}$$ ; hence hitting with the simple-current automorphism $$\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]$$ , we may assume\n$$\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 358, 542, 444], "content": "Next consider even $$r\\neq4$$ and odd $$k\\,>\\,2$$ . Either of $$\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}$$ or $$\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}$$ is\nimpossible, by comparing $$S_{\\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\\Lambda_{1},J0}$$ for any simple-current $$J$$ . For any of the\nthree remaining choices of $$J_{v}^{a}J_{s}^{b}\\Lambda_{1}$$ , we can find a simple-current automorphism of the form\n$$\\pi\\left[{\\ast}\\quad a\\,\\right];$$ hitting its inverse onto $$\\pi$$ allows us to take $$a=b=0$$ . Again $$0\\not=S_{\\Lambda_{1}\\Lambda_{1}}$$ forces\n$$b^{\\prime}=0$$ , and now $$a^{\\prime}=1$$ is forbidden. Thus again $$\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 444, 541, 488], "content": "As usual, $$r=4$$ is complicated by triality. We can force $$\\pi\\Lambda_{1}=\\Lambda_{1}$$ . That we can also\ntake $$\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$$ , follows from the inequality $$\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0$$ , valid for\n$$k\\geq3$$ . Establishing that inequality from (3.4) is equivalent to showing", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [98, 500, 514, 516], "content": "$$1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0$$ 2)", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [69, 528, 398, 542], "content": "for $$0<x\\le2\\pi/9$$ , which can be shown e.g. using Taylor series.", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [70, 543, 541, 657], "content": "For odd $$r$$ , the charge-conjugation $$C$$ equals $$C_{1}$$ . Since it must commute with $$\\pi$$ , i.e.\nthat $$C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}$$ must equal $$\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$$ , we get that $$b=0$$ . Similarly $$b^{\\prime}=0$$ .\nWhen $$k$$ is odd, eliminate $$a=1$$ and $$a\\prime=1$$ by comparing $$S_{\\Lambda_{1},J_{s}0}$$ and $$S_{J_{v}\\Lambda_{1},J0}$$ as before.\nThe hardest case is $$k$$ even. We can force $$\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}$$ by hitting with $$\\pi[a]$$ . Suppose for\ncontradiction that $$\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}$$ . We know $$\\pi^{\\prime}(J_{v}0)=J_{v}0$$ (compare $$S_{\\Lambda_{1},J_{v}0}$$ and $$S_{\\Lambda_{1},J0})$$ ,\nso by (2.7b) $$\\pi\\Lambda_{r}$$ must be a spinor. $$\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]$$ requires $$\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}$$ .\nFrom the $$\\Lambda_{1}\\boxtimes\\Lambda_{r}$$ fusion we get $$\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}$$ , but $$C\\pi=\\pi C$$ says that $$\\pi\\Lambda_{r-1}=$$\n$$C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}$$ \u2014 a contradiction.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [70, 658, 541, 715], "content": "Thus in all cases we have $$\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}$$ . We know $$\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0$$ (compare\n$$S_{\\Lambda_{1},J_{v}0}$$ and $$S_{\\Lambda_{1},J0})$$ , so $$\\pi\\Lambda_{r}$$ is a spinor and in fact must equal $$\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}$$ . Hitting\nwith $$(C_{1}^{i}\\pi_{v}^{j})^{-1}$$ , we can require $$\\pi\\Lambda_{r}=\\Lambda_{r}$$ . That $$\\pi\\Lambda_{r-1}$$ must now equal $$\\Lambda_{r-1}$$ follows from\nthe $$\\Lambda_{1}\\boxtimes\\Lambda_{r}$$ fusion.", "block_type": "text", "index": 13}]	[{"type": "inline_equation", "coordinates": [71, 75, 126, 84], "content": "a=a^{\\prime}=0", "score": 0.93, "index": 1}, {"type": "text", "coordinates": [126, 72, 178, 90], "content": " holds for ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [178, 75, 191, 84], "content": "k r", "score": 0.39, "index": 3}, {"type": "text", "coordinates": [191, 72, 312, 90], "content": " even. From the fusion ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [313, 75, 326, 86], "content": "\\Lambda_{1}", "score": 0.64, "index": 5}, {"type": "text", "coordinates": [327, 72, 344, 90], "content": " \u00d7 ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [344, 74, 358, 86], "content": "\\Lambda_{\\ell}", "score": 0.63, "index": 7}, {"type": "text", "coordinates": [358, 72, 398, 90], "content": "we get ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [398, 75, 529, 87], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [529, 72, 542, 90], "content": " if", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [71, 90, 120, 100], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [121, 89, 145, 102], "content": "; for ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [146, 90, 174, 99], "content": "r<k", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [174, 89, 408, 102], "content": " conclude the argument with the calculation", "score": 1.0, "index": 14}, {"type": "interline_equation", "coordinates": [99, 113, 500, 141], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [70, 151, 100, 168], "content": "as in ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [100, 153, 121, 165], "content": "\\S4.3", "score": 0.41, "index": 17}, {"type": "text", "coordinates": [122, 151, 165, 168], "content": ". When ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [165, 154, 195, 163], "content": "r=k", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [195, 151, 360, 168], "content": ", that inequality only holds for ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [360, 153, 389, 164], "content": "\\ell>1", "score": 0.88, "index": 21}, {"type": "text", "coordinates": [390, 151, 487, 168], "content": ", but we can force ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [488, 154, 539, 165], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [70, 166, 127, 183], "content": "by hitting ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [127, 172, 135, 178], "content": "\\pi", "score": 0.77, "index": 25}, {"type": "text", "coordinates": [135, 166, 229, 183], "content": " if necessary with ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [229, 172, 249, 179], "content": "\\pi_{\\mathrm{rld}}", "score": 0.88, "index": 27}, {"type": "text", "coordinates": [249, 166, 254, 183], "content": ".", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [93, 180, 203, 197], "content": "The remaining case ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [203, 182, 225, 195], "content": "C_{2,3}", "score": 0.91, "index": 30}, {"type": "text", "coordinates": [225, 180, 314, 197], "content": " follows because ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [314, 182, 371, 192], "content": "\\pi^{\\prime}J0\\,=\\,J0", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [372, 180, 434, 197], "content": ": by (2.7b) ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [434, 182, 495, 194], "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "score": 0.87, "index": 34}, {"type": "text", "coordinates": [495, 180, 541, 197], "content": ", and by", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [72, 196, 105, 210], "content": "(2.7a) ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [105, 195, 162, 208], "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [163, 196, 170, 210], "content": " (", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [171, 195, 191, 208], "content": "3\\Lambda_{1}", "score": 0.88, "index": 39}, {"type": "text", "coordinates": [191, 196, 216, 210], "content": " is a ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [216, 196, 225, 206], "content": "J", "score": 0.84, "index": 41}, {"type": "text", "coordinates": [225, 196, 292, 210], "content": "-fixed-point).", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [71, 225, 119, 236], "content": "4.5. The ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [119, 225, 130, 235], "content": "D", "score": 0.8, "index": 44}, {"type": "text", "coordinates": [131, 225, 219, 236], "content": "-series argument", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [95, 247, 124, 256], "content": "k=1", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [124, 244, 201, 259], "content": " is trivial, and ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [202, 245, 231, 256], "content": "k=2", "score": 0.88, "index": 48}, {"type": "text", "coordinates": [232, 244, 398, 259], "content": " will be considered shortly. For ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [398, 247, 427, 256], "content": "k>2", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [427, 244, 541, 259], "content": ", Proposition 4.1 tells", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [70, 258, 111, 274], "content": "us that", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [112, 259, 189, 273], "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "score": 0.93, "index": 53}, {"type": "text", "coordinates": [190, 258, 216, 274], "content": " and ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [216, 258, 303, 273], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "score": 0.94, "index": 55}, {"type": "text", "coordinates": [303, 258, 329, 274], "content": ", for ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [329, 260, 419, 273], "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [419, 258, 541, 274], "content": ". Immediate from (3.4)", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [69, 272, 109, 289], "content": "is that ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [110, 274, 173, 288], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "score": 0.92, "index": 60}, {"type": "text", "coordinates": [173, 272, 226, 289], "content": " and that ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [226, 275, 261, 287], "content": "\\chi_{\\Lambda_{1}}[\\psi]", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [262, 272, 334, 289], "content": ", for a spinor ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [334, 275, 343, 287], "content": "\\psi", "score": 0.89, "index": 64}, {"type": "text", "coordinates": [343, 272, 469, 289], "content": ", takes its maximum at ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [469, 274, 509, 287], "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [509, 272, 542, 289], "content": ". Our", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [70, 288, 176, 303], "content": "first step is to force ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [177, 289, 268, 301], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [268, 288, 502, 303], "content": ". Unfortunately this requires a case analysis.", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [94, 301, 201, 319], "content": "Consider first even ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [201, 304, 235, 315], "content": "r\\,\\neq\\,4", "score": 0.92, "index": 72}, {"type": "text", "coordinates": [235, 301, 297, 319], "content": ", and even ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [297, 304, 331, 313], "content": "k\\ >\\ 2", "score": 0.88, "index": 74}, {"type": "text", "coordinates": [332, 301, 376, 319], "content": ". Now, ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [377, 304, 504, 316], "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [504, 301, 542, 319], "content": " forces", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [71, 325, 104, 335], "content": "b=b^{\\prime}", "score": 0.9, "index": 78}, {"type": "text", "coordinates": [104, 324, 394, 339], "content": "; hence hitting with the simple-current automorphism ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [394, 317, 449, 347], "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "score": 0.94, "index": 80}, {"type": "text", "coordinates": [450, 326, 541, 338], "content": ", we may assume", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [71, 347, 162, 358], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [163, 347, 165, 361], "content": ".", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [93, 359, 200, 375], "content": "Next consider even ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [200, 362, 230, 373], "content": "r\\neq4", "score": 0.93, "index": 85}, {"type": "text", "coordinates": [230, 359, 281, 375], "content": " and odd ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [282, 362, 312, 371], "content": "k\\,>\\,2", "score": 0.9, "index": 87}, {"type": "text", "coordinates": [313, 359, 374, 375], "content": ". Either of ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [374, 362, 439, 373], "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "score": 0.92, "index": 89}, {"type": "text", "coordinates": [439, 359, 458, 375], "content": " or ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [459, 360, 527, 373], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "score": 0.92, "index": 91}, {"type": "text", "coordinates": [527, 359, 541, 375], "content": " is", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [69, 373, 210, 392], "content": "impossible, by comparing ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [210, 376, 246, 388], "content": "S_{\\Lambda_{1},J_{s}0}", "score": 0.92, "index": 94}, {"type": "text", "coordinates": [247, 373, 275, 392], "content": " and ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [275, 376, 318, 388], "content": "S_{J_{v}\\Lambda_{1},J0}", "score": 0.93, "index": 96}, {"type": "text", "coordinates": [319, 373, 444, 392], "content": " for any simple-current ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [444, 376, 453, 385], "content": "J", "score": 0.86, "index": 98}, {"type": "text", "coordinates": [453, 373, 542, 392], "content": ". For any of the", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [70, 387, 208, 405], "content": "three remaining choices of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [209, 389, 249, 402], "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "score": 0.93, "index": 101}, {"type": "text", "coordinates": [249, 387, 541, 405], "content": ", we can find a simple-current automorphism of the form", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [71, 403, 123, 433], "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "score": 0.95, "index": 103}, {"type": "text", "coordinates": [124, 409, 252, 425], "content": " hitting its inverse onto ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [253, 415, 260, 421], "content": "\\pi", "score": 0.88, "index": 105}, {"type": "text", "coordinates": [260, 409, 356, 425], "content": " allows us to take ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [357, 412, 408, 421], "content": "a=b=0", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [408, 409, 452, 425], "content": ". Again ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [452, 412, 505, 424], "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "score": 0.95, "index": 109}, {"type": "text", "coordinates": [505, 409, 541, 425], "content": " forces", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [71, 433, 102, 443], "content": "b^{\\prime}=0", "score": 0.87, "index": 111}, {"type": "text", "coordinates": [102, 431, 157, 447], "content": ", and now ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [157, 433, 189, 443], "content": "a^{\\prime}=1", "score": 0.91, "index": 113}, {"type": "text", "coordinates": [189, 431, 325, 447], "content": " is forbidden. Thus again ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [325, 433, 416, 444], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [416, 431, 420, 447], "content": ".", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [95, 446, 147, 461], "content": "As usual, ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [147, 448, 176, 457], "content": "r=4", "score": 0.9, "index": 118}, {"type": "text", "coordinates": [176, 446, 389, 461], "content": " is complicated by triality. We can force ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [390, 448, 441, 459], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [441, 446, 541, 461], "content": ". That we can also", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [70, 460, 97, 477], "content": "take ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [97, 461, 154, 473], "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "score": 0.92, "index": 123}, {"type": "text", "coordinates": [154, 460, 307, 477], "content": ", follows from the inequality ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [307, 461, 488, 474], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "score": 0.92, "index": 125}, {"type": "text", "coordinates": [488, 460, 541, 477], "content": ", valid for", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [71, 476, 100, 487], "content": "k\\geq3", "score": 0.89, "index": 127}, {"type": "text", "coordinates": [100, 473, 442, 492], "content": ". Establishing that inequality from (3.4) is equivalent to showing", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [98, 503, 500, 516], "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "score": 0.77, "index": 129}, {"type": "text", "coordinates": [501, 502, 512, 520], "content": "2)", "score": 1.0, "index": 130}, {"type": "text", "coordinates": [70, 530, 89, 545], "content": "for ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [89, 531, 160, 544], "content": "0<x\\le2\\pi/9", "score": 0.92, "index": 132}, {"type": "text", "coordinates": [160, 530, 397, 545], "content": ", which can be shown e.g. using Taylor series.", "score": 1.0, "index": 133}, {"type": "text", "coordinates": [93, 543, 141, 560], "content": "For odd ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [141, 550, 147, 556], "content": "r", "score": 0.8, "index": 135}, {"type": "text", "coordinates": [147, 543, 279, 560], "content": ", the charge-conjugation ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [279, 547, 289, 556], "content": "C", "score": 0.88, "index": 137}, {"type": "text", "coordinates": [289, 543, 329, 560], "content": " equals ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [330, 547, 344, 558], "content": "C_{1}", "score": 0.91, "index": 139}, {"type": "text", "coordinates": [344, 543, 508, 560], "content": ". Since it must commute with ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [509, 550, 516, 556], "content": "\\pi", "score": 0.86, "index": 141}, {"type": "text", "coordinates": [516, 543, 540, 560], "content": ", i.e.", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [69, 556, 96, 574], "content": "that ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [96, 558, 198, 573], "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "score": 0.93, "index": 144}, {"type": "text", "coordinates": [199, 556, 261, 574], "content": " must equal ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [262, 559, 352, 573], "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "score": 0.95, "index": 146}, {"type": "text", "coordinates": [353, 556, 419, 574], "content": ", we get that", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [420, 561, 447, 570], "content": "b=0", "score": 0.9, "index": 148}, {"type": "text", "coordinates": [448, 556, 505, 574], "content": ". Similarly ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [506, 560, 536, 570], "content": "b^{\\prime}=0", "score": 0.92, "index": 150}, {"type": "text", "coordinates": [537, 556, 541, 574], "content": ".", "score": 1.0, "index": 151}, {"type": "text", "coordinates": [70, 572, 106, 590], "content": "When ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [106, 574, 113, 584], "content": "k", "score": 0.78, "index": 153}, {"type": "text", "coordinates": [114, 572, 208, 590], "content": " is odd, eliminate ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [208, 574, 238, 584], "content": "a=1", "score": 0.86, "index": 155}, {"type": "text", "coordinates": [238, 572, 265, 590], "content": " and ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [265, 574, 298, 584], "content": "a\\prime=1", "score": 0.92, "index": 157}, {"type": "text", "coordinates": [298, 572, 377, 590], "content": " by comparing ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [377, 575, 414, 587], "content": "S_{\\Lambda_{1},J_{s}0}", "score": 0.92, "index": 159}, {"type": "text", "coordinates": [414, 572, 441, 590], "content": " and ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [442, 575, 485, 587], "content": "S_{J_{v}\\Lambda_{1},J0}", "score": 0.93, "index": 161}, {"type": "text", "coordinates": [485, 572, 541, 590], "content": " as before.", "score": 1.0, "index": 162}, {"type": "text", "coordinates": [70, 587, 179, 602], "content": "The hardest case is ", "score": 1.0, "index": 163}, {"type": "inline_equation", "coordinates": [180, 588, 187, 599], "content": "k", "score": 0.85, "index": 164}, {"type": "text", "coordinates": [187, 587, 300, 602], "content": " even. We can force ", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [300, 590, 354, 600], "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "score": 0.92, "index": 166}, {"type": "text", "coordinates": [354, 587, 445, 602], "content": " by hitting with ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [445, 589, 466, 601], "content": "\\pi[a]", "score": 0.92, "index": 168}, {"type": "text", "coordinates": [466, 587, 541, 602], "content": ". Suppose for", "score": 1.0, "index": 169}, {"type": "text", "coordinates": [70, 601, 170, 617], "content": "contradiction that ", "score": 1.0, "index": 170}, {"type": "inline_equation", "coordinates": [171, 601, 239, 614], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "score": 0.92, "index": 171}, {"type": "text", "coordinates": [239, 601, 301, 617], "content": ". We know ", "score": 1.0, "index": 172}, {"type": "inline_equation", "coordinates": [302, 603, 376, 616], "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "score": 0.93, "index": 173}, {"type": "text", "coordinates": [376, 601, 433, 617], "content": " (compare ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [433, 604, 470, 616], "content": "S_{\\Lambda_{1},J_{v}0}", "score": 0.93, "index": 175}, {"type": "text", "coordinates": [470, 601, 498, 617], "content": " and ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [498, 603, 536, 616], "content": "S_{\\Lambda_{1},J0})", "score": 0.89, "index": 177}, {"type": "text", "coordinates": [537, 601, 541, 617], "content": ",", "score": 1.0, "index": 178}, {"type": "text", "coordinates": [69, 615, 141, 633], "content": "so by (2.7b) ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [141, 616, 163, 629], "content": "\\pi\\Lambda_{r}", "score": 0.86, "index": 180}, {"type": "text", "coordinates": [163, 615, 272, 633], "content": " must be a spinor.", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [273, 617, 390, 630], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "score": 0.92, "index": 182}, {"type": "text", "coordinates": [391, 615, 442, 633], "content": " requires ", "score": 1.0, "index": 183}, {"type": "inline_equation", "coordinates": [442, 617, 536, 630], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "score": 0.93, "index": 184}, {"type": "text", "coordinates": [537, 615, 540, 633], "content": ".", "score": 1.0, "index": 185}, {"type": "text", "coordinates": [69, 630, 123, 646], "content": "From the ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [123, 631, 169, 644], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "score": 0.52, "index": 187}, {"type": "text", "coordinates": [169, 630, 245, 646], "content": " fusion we get ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [245, 631, 360, 644], "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "score": 0.92, "index": 189}, {"type": "text", "coordinates": [360, 630, 388, 646], "content": ", but ", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [389, 633, 438, 641], "content": "C\\pi=\\pi C", "score": 0.92, "index": 191}, {"type": "text", "coordinates": [438, 630, 493, 646], "content": " says that ", "score": 1.0, "index": 192}, {"type": "inline_equation", "coordinates": [493, 632, 541, 644], "content": "\\pi\\Lambda_{r-1}=", "score": 0.9, "index": 193}, {"type": "inline_equation", "coordinates": [71, 644, 147, 659], "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "score": 0.91, "index": 194}, {"type": "text", "coordinates": [148, 644, 251, 662], "content": " \u2014 a contradiction.", "score": 1.0, "index": 195}, {"type": "text", "coordinates": [93, 658, 239, 675], "content": "Thus in all cases we have ", "score": 1.0, "index": 196}, {"type": "inline_equation", "coordinates": [240, 660, 340, 672], "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "score": 0.92, "index": 197}, {"type": "text", "coordinates": [341, 658, 407, 675], "content": ". We know ", "score": 1.0, "index": 198}, {"type": "inline_equation", "coordinates": [408, 660, 486, 673], "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "score": 0.93, "index": 199}, {"type": "text", "coordinates": [486, 658, 541, 675], "content": " (compare", "score": 1.0, "index": 200}, {"type": "inline_equation", "coordinates": [71, 673, 109, 687], "content": "S_{\\Lambda_{1},J_{v}0}", "score": 0.91, "index": 201}, {"type": "text", "coordinates": [109, 674, 137, 689], "content": " and ", "score": 1.0, "index": 202}, {"type": "inline_equation", "coordinates": [137, 673, 175, 687], "content": "S_{\\Lambda_{1},J0})", "score": 0.89, "index": 203}, {"type": "text", "coordinates": [175, 674, 198, 689], "content": ", so ", "score": 1.0, "index": 204}, {"type": "inline_equation", "coordinates": [198, 675, 219, 686], "content": "\\pi\\Lambda_{r}", "score": 0.9, "index": 205}, {"type": "text", "coordinates": [220, 674, 410, 689], "content": " is a spinor and in fact must equal ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [411, 674, 491, 687], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "score": 0.93, "index": 207}, {"type": "text", "coordinates": [491, 674, 541, 689], "content": ". Hitting", "score": 1.0, "index": 208}, {"type": "text", "coordinates": [70, 687, 98, 704], "content": "with ", "score": 1.0, "index": 209}, {"type": "inline_equation", "coordinates": [99, 687, 147, 702], "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "score": 0.93, "index": 210}, {"type": "text", "coordinates": [147, 687, 234, 704], "content": ", we can require ", "score": 1.0, "index": 211}, {"type": "inline_equation", "coordinates": [234, 690, 285, 700], "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "score": 0.91, "index": 212}, {"type": "text", "coordinates": [285, 687, 322, 704], "content": ". That ", "score": 1.0, "index": 213}, {"type": "inline_equation", "coordinates": [323, 690, 356, 701], "content": "\\pi\\Lambda_{r-1}", "score": 0.92, "index": 214}, {"type": "text", "coordinates": [356, 687, 445, 704], "content": " must now equal ", "score": 1.0, "index": 215}, {"type": "inline_equation", "coordinates": [446, 690, 471, 701], "content": "\\Lambda_{r-1}", "score": 0.92, "index": 216}, {"type": "text", "coordinates": [472, 687, 542, 704], "content": " follows from", "score": 1.0, "index": 217}, {"type": "text", "coordinates": [71, 703, 91, 717], "content": "the ", "score": 1.0, "index": 218}, {"type": "inline_equation", "coordinates": [92, 703, 137, 715], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "score": 0.28, "index": 219}, {"type": "text", "coordinates": [138, 703, 177, 717], "content": " fusion.", "score": 1.0, "index": 220}]	[]	[{"type": "block", "coordinates": [99, 113, 500, 141], "content": "", "caption": ""}, {"type": "inline", "coordinates": [71, 75, 126, 84], "content": "a=a^{\\prime}=0", "caption": ""}, {"type": "inline", "coordinates": [178, 75, 191, 84], "content": "k r", "caption": ""}, {"type": "inline", "coordinates": [313, 75, 326, 86], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [344, 74, 358, 86], "content": "\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [398, 75, 529, 87], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "caption": ""}, {"type": "inline", "coordinates": [71, 90, 120, 100], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [146, 90, 174, 99], "content": "r<k", "caption": ""}, {"type": "inline", "coordinates": [100, 153, 121, 165], "content": "\\S4.3", "caption": ""}, {"type": "inline", "coordinates": [165, 154, 195, 163], "content": "r=k", "caption": ""}, {"type": "inline", "coordinates": [360, 153, 389, 164], "content": "\\ell>1", "caption": ""}, {"type": "inline", "coordinates": [488, 154, 539, 165], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [127, 172, 135, 178], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [229, 172, 249, 179], "content": "\\pi_{\\mathrm{rld}}", "caption": ""}, {"type": "inline", "coordinates": [203, 182, 225, 195], "content": "C_{2,3}", "caption": ""}, {"type": "inline", "coordinates": [314, 182, 371, 192], "content": "\\pi^{\\prime}J0\\,=\\,J0", "caption": ""}, {"type": "inline", "coordinates": [434, 182, 495, 194], "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "caption": ""}, {"type": "inline", "coordinates": [105, 195, 162, 208], "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [171, 195, 191, 208], "content": "3\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [216, 196, 225, 206], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [119, 225, 130, 235], "content": "D", "caption": ""}, {"type": "inline", "coordinates": [95, 247, 124, 256], "content": "k=1", "caption": ""}, {"type": "inline", "coordinates": [202, 245, 231, 256], "content": "k=2", "caption": ""}, {"type": "inline", "coordinates": [398, 247, 427, 256], "content": "k>2", "caption": ""}, {"type": "inline", "coordinates": [112, 259, 189, 273], "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [216, 258, 303, 273], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [329, 260, 419, 273], "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "caption": ""}, {"type": "inline", "coordinates": [110, 274, 173, 288], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "caption": ""}, {"type": "inline", "coordinates": [226, 275, 261, 287], "content": "\\chi_{\\Lambda_{1}}[\\psi]", "caption": ""}, {"type": "inline", "coordinates": [334, 275, 343, 287], "content": "\\psi", "caption": ""}, {"type": "inline", "coordinates": [469, 274, 509, 287], "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [177, 289, 268, 301], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [201, 304, 235, 315], "content": "r\\,\\neq\\,4", "caption": ""}, {"type": "inline", "coordinates": [297, 304, 331, 313], "content": "k\\ >\\ 2", "caption": ""}, {"type": "inline", "coordinates": [377, 304, 504, 316], "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "caption": ""}, {"type": "inline", "coordinates": [71, 325, 104, 335], "content": "b=b^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [394, 317, 449, 347], "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "caption": ""}, {"type": "inline", "coordinates": [71, 347, 162, 358], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [200, 362, 230, 373], "content": "r\\neq4", "caption": ""}, {"type": "inline", "coordinates": [282, 362, 312, 371], "content": "k\\,>\\,2", "caption": ""}, {"type": "inline", "coordinates": [374, 362, 439, 373], "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [459, 360, 527, 373], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [210, 376, 246, 388], "content": "S_{\\Lambda_{1},J_{s}0}", "caption": ""}, {"type": "inline", "coordinates": [275, 376, 318, 388], "content": "S_{J_{v}\\Lambda_{1},J0}", "caption": ""}, {"type": "inline", "coordinates": [444, 376, 453, 385], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [209, 389, 249, 402], "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [71, 403, 123, 433], "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "caption": ""}, {"type": "inline", "coordinates": [253, 415, 260, 421], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [357, 412, 408, 421], "content": "a=b=0", "caption": ""}, {"type": "inline", "coordinates": [452, 412, 505, 424], "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "caption": ""}, {"type": "inline", "coordinates": [71, 433, 102, 443], "content": "b^{\\prime}=0", "caption": ""}, {"type": "inline", "coordinates": [157, 433, 189, 443], "content": "a^{\\prime}=1", "caption": ""}, {"type": "inline", "coordinates": [325, 433, 416, 444], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [147, 448, 176, 457], "content": "r=4", "caption": ""}, {"type": "inline", "coordinates": [390, 448, 441, 459], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [97, 461, 154, 473], "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [307, 461, 488, 474], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "caption": ""}, {"type": "inline", "coordinates": [71, 476, 100, 487], "content": "k\\geq3", "caption": ""}, {"type": "inline", "coordinates": [98, 503, 500, 516], "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "caption": ""}, {"type": "inline", "coordinates": [89, 531, 160, 544], "content": "0<x\\le2\\pi/9", "caption": ""}, {"type": "inline", "coordinates": [141, 550, 147, 556], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [279, 547, 289, 556], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [330, 547, 344, 558], "content": "C_{1}", "caption": ""}, {"type": "inline", "coordinates": [509, 550, 516, 556], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [96, 558, 198, 573], "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [262, 559, 352, 573], "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [420, 561, 447, 570], "content": "b=0", "caption": ""}, {"type": "inline", "coordinates": [506, 560, 536, 570], "content": "b^{\\prime}=0", "caption": ""}, {"type": "inline", "coordinates": [106, 574, 113, 584], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [208, 574, 238, 584], "content": "a=1", "caption": ""}, {"type": "inline", "coordinates": [265, 574, 298, 584], "content": "a\\prime=1", "caption": ""}, {"type": "inline", "coordinates": [377, 575, 414, 587], "content": "S_{\\Lambda_{1},J_{s}0}", "caption": ""}, {"type": "inline", "coordinates": [442, 575, 485, 587], "content": "S_{J_{v}\\Lambda_{1},J0}", "caption": ""}, {"type": "inline", "coordinates": [180, 588, 187, 599], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [300, 590, 354, 600], "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [445, 589, 466, 601], "content": "\\pi[a]", "caption": ""}, {"type": "inline", "coordinates": [171, 601, 239, 614], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [302, 603, 376, 616], "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "caption": ""}, {"type": "inline", "coordinates": [433, 604, 470, 616], "content": "S_{\\Lambda_{1},J_{v}0}", "caption": ""}, {"type": "inline", "coordinates": [498, 603, 536, 616], "content": "S_{\\Lambda_{1},J0})", "caption": ""}, {"type": "inline", "coordinates": [141, 616, 163, 629], "content": "\\pi\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [273, 617, 390, 630], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "caption": ""}, {"type": "inline", "coordinates": [442, 617, 536, 630], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [123, 631, 169, 644], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [245, 631, 360, 644], "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "caption": ""}, {"type": "inline", "coordinates": [389, 633, 438, 641], "content": "C\\pi=\\pi C", "caption": ""}, {"type": "inline", "coordinates": [493, 632, 541, 644], "content": "\\pi\\Lambda_{r-1}=", "caption": ""}, {"type": "inline", "coordinates": [71, 644, 147, 659], "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "caption": ""}, {"type": "inline", "coordinates": [240, 660, 340, 672], "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [408, 660, 486, 673], "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "caption": ""}, {"type": "inline", "coordinates": [71, 673, 109, 687], "content": "S_{\\Lambda_{1},J_{v}0}", "caption": ""}, {"type": "inline", "coordinates": [137, 673, 175, 687], "content": "S_{\\Lambda_{1},J0})", "caption": ""}, {"type": "inline", "coordinates": [198, 675, 219, 686], "content": "\\pi\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [411, 674, 491, 687], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [99, 687, 147, 702], "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "caption": ""}, {"type": "inline", "coordinates": [234, 690, 285, 700], "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [323, 690, 356, 701], "content": "\\pi\\Lambda_{r-1}", "caption": ""}, {"type": "inline", "coordinates": [446, 690, 471, 701], "content": "\\Lambda_{r-1}", "caption": ""}, {"type": "inline", "coordinates": [92, 703, 137, 715], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 18}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>\n$$", "text_format": "latex", "page_idx": 18}, {"type": "text", "text": "as in $\\S4.3$ . When $r=k$ , that inequality only holds for $\\ell>1$ , but we can force $\\pi\\Lambda_{2}=\\Lambda_{2}$ by hitting $\\pi$ if necessary with $\\pi_{\\mathrm{rld}}$ . ", "page_idx": 18}, {"type": "text", "text": "The remaining case $C_{2,3}$ follows because $\\pi^{\\prime}J0\\,=\\,J0$ : by (2.7b) $\\pi\\Lambda_{1}\\notin S\\Lambda_{2}$ , and by (2.7a) $\\pi\\Lambda_{1}\\neq3\\Lambda_{1}$ ( $3\\Lambda_{1}$ is a $J$ -fixed-point). ", "page_idx": 18}, {"type": "text", "text": "4.5. The $D$ -series argument ", "text_level": 1, "page_idx": 18}, {"type": "text", "text": "$k=1$ is trivial, and $k=2$ will be considered shortly. For $k>2$ , Proposition 4.1 tells us that $\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ and $\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}$ , for $a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}$ . Immediate from (3.4) is that $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$ and that $\\chi_{\\Lambda_{1}}[\\psi]$ , for a spinor $\\psi$ , takes its maximum at $C^{i}J_{v}^{j}\\Lambda_{r}$ . Our first step is to force $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . Unfortunately this requires a case analysis. ", "page_idx": 18}, {"type": "text", "text": "Consider first even $r\\,\\neq\\,4$ , and even $k\\ >\\ 2$ . Now, $0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}$ forces $b=b^{\\prime}$ ; hence hitting with the simple-current automorphism $\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]$ , we may assume $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 18}, {"type": "text", "text": "Next consider even $r\\neq4$ and odd $k\\,>\\,2$ . Either of $\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}$ or $\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}$ is impossible, by comparing $S_{\\Lambda_{1},J_{s}0}$ and $S_{J_{v}\\Lambda_{1},J0}$ for any simple-current $J$ . For any of the three remaining choices of $J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ , we can find a simple-current automorphism of the form $\\pi\\left[{\\ast}\\quad a\\,\\right];$ hitting its inverse onto $\\pi$ allows us to take $a=b=0$ . Again $0\\not=S_{\\Lambda_{1}\\Lambda_{1}}$ forces $b^{\\prime}=0$ , and now $a^{\\prime}=1$ is forbidden. Thus again $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 18}, {"type": "text", "text": "As usual, $r=4$ is complicated by triality. We can force $\\pi\\Lambda_{1}=\\Lambda_{1}$ . That we can also take $\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ , follows from the inequality $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0$ , valid for $k\\geq3$ . Establishing that inequality from (3.4) is equivalent to showing ", "page_idx": 18}, {"type": "text", "text": "$1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0$ 2) ", "page_idx": 18}, {"type": "text", "text": "for $0<x\\le2\\pi/9$ , which can be shown e.g. using Taylor series. ", "page_idx": 18}, {"type": "text", "text": "For odd $r$ , the charge-conjugation $C$ equals $C_{1}$ . Since it must commute with $\\pi$ , i.e. that $C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}$ must equal $\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ , we get that $b=0$ . Similarly $b^{\\prime}=0$ . When $k$ is odd, eliminate $a=1$ and $a\\prime=1$ by comparing $S_{\\Lambda_{1},J_{s}0}$ and $S_{J_{v}\\Lambda_{1},J0}$ as before. The hardest case is $k$ even. We can force $\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}$ by hitting with $\\pi[a]$ . Suppose for contradiction that $\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}$ . We know $\\pi^{\\prime}(J_{v}0)=J_{v}0$ (compare $S_{\\Lambda_{1},J_{v}0}$ and $S_{\\Lambda_{1},J0})$ , so by (2.7b) $\\pi\\Lambda_{r}$ must be a spinor. $\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]$ requires $\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}$ . From the $\\Lambda_{1}\\boxtimes\\Lambda_{r}$ fusion we get $\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}$ , but $C\\pi=\\pi C$ says that $\\pi\\Lambda_{r-1}=$ $C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}$ \u2014 a contradiction. ", "page_idx": 18}, {"type": "text", "text": "Thus in all cases we have $\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}$ . We know $\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0$ (compare $S_{\\Lambda_{1},J_{v}0}$ and $S_{\\Lambda_{1},J0})$ , so $\\pi\\Lambda_{r}$ is a spinor and in fact must equal $\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}$ . Hitting with $(C_{1}^{i}\\pi_{v}^{j})^{-1}$ , we can require $\\pi\\Lambda_{r}=\\Lambda_{r}$ . That $\\pi\\Lambda_{r-1}$ must now equal $\\Lambda_{r-1}$ follows from the $\\Lambda_{1}\\boxtimes\\Lambda_{r}$ fusion. 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{"category_id": 15, "poly": [533.0, 545.0, 601.0, 545.0, 601.0, 584.0, 533.0, 584.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [627.0, 545.0, 813.0, 545.0, 813.0, 584.0, 627.0, 584.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [197.0, 1474.0, 248.0, 1474.0, 248.0, 1515.0, 197.0, 1515.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [446.0, 1474.0, 1104.0, 1474.0, 1104.0, 1515.0, 446.0, 1515.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [829.0, 2033.0, 871.0, 2033.0, 871.0, 2070.0, 829.0, 2070.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [199.0, 625.0, 331.0, 625.0, 331.0, 657.0, 199.0, 657.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [364.0, 625.0, 609.0, 625.0, 609.0, 657.0, 364.0, 657.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1392.0, 1395.0, 1423.0, 1395.0, 1423.0, 1445.0, 1392.0, 1445.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [69, 70, 542, 100], "lines": [{"bbox": [71, 72, 542, 90], "spans": [{"bbox": [71, 75, 126, 84], "score": 0.93, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 9, "width": 55}, {"bbox": [126, 72, 178, 90], "score": 1.0, "content": " holds for ", "type": "text"}, {"bbox": [178, 75, 191, 84], "score": 0.39, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [191, 72, 312, 90], "score": 1.0, "content": " even. From the fusion ", "type": "text"}, {"bbox": [313, 75, 326, 86], "score": 0.64, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [327, 72, 344, 90], "score": 1.0, "content": " \u00d7 ", "type": "text"}, {"bbox": [344, 74, 358, 86], "score": 0.63, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [358, 72, 398, 90], "score": 1.0, "content": "we get ", "type": "text"}, {"bbox": [398, 75, 529, 87], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 12, "width": 131}, {"bbox": [529, 72, 542, 90], "score": 1.0, "content": " if", "type": "text"}], "index": 0}, {"bbox": [71, 89, 408, 102], "spans": [{"bbox": [71, 90, 120, 100], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [121, 89, 145, 102], "score": 1.0, "content": "; for ", "type": "text"}, {"bbox": [146, 90, 174, 99], "score": 0.92, "content": "r<k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [174, 89, 408, 102], "score": 1.0, "content": " conclude the argument with the calculation", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [70, 149, 541, 178], "lines": [{"bbox": [70, 151, 539, 168], "spans": [{"bbox": [70, 151, 100, 168], "score": 1.0, "content": "as in ", "type": "text"}, {"bbox": [100, 153, 121, 165], "score": 0.41, "content": "\\S4.3", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [122, 151, 165, 168], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [165, 154, 195, 163], "score": 0.92, "content": "r=k", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [195, 151, 360, 168], "score": 1.0, "content": ", that inequality only holds for ", "type": "text"}, {"bbox": [360, 153, 389, 164], "score": 0.88, "content": "\\ell>1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [390, 151, 487, 168], "score": 1.0, "content": ", but we can force ", "type": "text"}, {"bbox": [488, 154, 539, 165], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 51}], "index": 3}, {"bbox": [70, 166, 254, 183], "spans": [{"bbox": [70, 166, 127, 183], "score": 1.0, "content": "by hitting ", "type": "text"}, {"bbox": [127, 172, 135, 178], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [135, 166, 229, 183], "score": 1.0, "content": " if necessary with ", "type": "text"}, {"bbox": [229, 172, 249, 179], "score": 0.88, "content": "\\pi_{\\mathrm{rld}}", "type": "inline_equation", "height": 7, "width": 20}, {"bbox": [249, 166, 254, 183], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [70, 179, 541, 208], "lines": [{"bbox": [93, 180, 541, 197], "spans": [{"bbox": [93, 180, 203, 197], "score": 1.0, "content": "The remaining case ", "type": "text"}, {"bbox": [203, 182, 225, 195], "score": 0.91, "content": "C_{2,3}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [225, 180, 314, 197], "score": 1.0, "content": " follows because ", "type": "text"}, {"bbox": [314, 182, 371, 192], "score": 0.92, "content": "\\pi^{\\prime}J0\\,=\\,J0", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [372, 180, 434, 197], "score": 1.0, "content": ": by (2.7b) ", "type": "text"}, {"bbox": [434, 182, 495, 194], "score": 0.87, "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [495, 180, 541, 197], "score": 1.0, "content": ", and by", "type": "text"}], "index": 5}, {"bbox": [72, 195, 292, 210], "spans": [{"bbox": [72, 196, 105, 210], "score": 1.0, "content": "(2.7a) ", "type": "text"}, {"bbox": [105, 195, 162, 208], "score": 0.89, "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [163, 196, 170, 210], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [171, 195, 191, 208], "score": 0.88, "content": "3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [191, 196, 216, 210], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [216, 196, 225, 206], "score": 0.84, "content": "J", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [225, 196, 292, 210], "score": 1.0, "content": "-fixed-point).", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "title", "bbox": [71, 221, 219, 236], "lines": [{"bbox": [71, 225, 219, 236], "spans": [{"bbox": [71, 225, 119, 236], "score": 1.0, "content": "4.5. The ", "type": "text"}, {"bbox": [119, 225, 130, 235], "score": 0.8, "content": "D", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [131, 225, 219, 236], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 243, 541, 300], "lines": [{"bbox": [95, 244, 541, 259], "spans": [{"bbox": [95, 247, 124, 256], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 244, 201, 259], "score": 1.0, "content": " is trivial, and ", "type": "text"}, {"bbox": [202, 245, 231, 256], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [232, 244, 398, 259], "score": 1.0, "content": " will be considered shortly. For ", "type": "text"}, {"bbox": [398, 247, 427, 256], "score": 0.9, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [427, 244, 541, 259], "score": 1.0, "content": ", Proposition 4.1 tells", "type": "text"}], "index": 8}, {"bbox": [70, 258, 541, 274], "spans": [{"bbox": [70, 258, 111, 274], "score": 1.0, "content": "us that", "type": "text"}, {"bbox": [112, 259, 189, 273], "score": 0.93, "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 77}, {"bbox": [190, 258, 216, 274], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [216, 258, 303, 273], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [303, 258, 329, 274], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [329, 260, 419, 273], "score": 0.93, "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [419, 258, 541, 274], "score": 1.0, "content": ". Immediate from (3.4)", "type": "text"}], "index": 9}, {"bbox": [69, 272, 542, 289], "spans": [{"bbox": [69, 272, 109, 289], "score": 1.0, "content": "is that ", "type": "text"}, {"bbox": [110, 274, 173, 288], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [173, 272, 226, 289], "score": 1.0, "content": " and that ", "type": "text"}, {"bbox": [226, 275, 261, 287], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\psi]", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [262, 272, 334, 289], "score": 1.0, "content": ", for a spinor ", "type": "text"}, {"bbox": [334, 275, 343, 287], "score": 0.89, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [343, 272, 469, 289], "score": 1.0, "content": ", takes its maximum at ", "type": "text"}, {"bbox": [469, 274, 509, 287], "score": 0.93, "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [509, 272, 542, 289], "score": 1.0, "content": ". Our", "type": "text"}], "index": 10}, {"bbox": [70, 288, 502, 303], "spans": [{"bbox": [70, 288, 176, 303], "score": 1.0, "content": "first step is to force ", "type": "text"}, {"bbox": [177, 289, 268, 301], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [268, 288, 502, 303], "score": 1.0, "content": ". Unfortunately this requires a case analysis.", "type": "text"}], "index": 11}], "index": 9.5}, {"type": "text", "bbox": [69, 301, 541, 358], "lines": [{"bbox": [94, 301, 542, 319], "spans": [{"bbox": [94, 301, 201, 319], "score": 1.0, "content": "Consider first even ", "type": "text"}, {"bbox": [201, 304, 235, 315], "score": 0.92, "content": "r\\,\\neq\\,4", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [235, 301, 297, 319], "score": 1.0, "content": ", and even ", "type": "text"}, {"bbox": [297, 304, 331, 313], "score": 0.88, "content": "k\\ >\\ 2", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [332, 301, 376, 319], "score": 1.0, "content": ". Now, ", "type": "text"}, {"bbox": [377, 304, 504, 316], "score": 0.92, "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 127}, {"bbox": [504, 301, 542, 319], "score": 1.0, "content": " forces", "type": "text"}], "index": 12}, {"bbox": [71, 317, 541, 347], "spans": [{"bbox": [71, 325, 104, 335], "score": 0.9, "content": "b=b^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [104, 324, 394, 339], "score": 1.0, "content": "; hence hitting with the simple-current automorphism ", "type": "text"}, {"bbox": [394, 317, 449, 347], "score": 0.94, "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [450, 326, 541, 338], "score": 1.0, "content": ", we may assume", "type": "text"}], "index": 13}, {"bbox": [71, 347, 165, 361], "spans": [{"bbox": [71, 347, 162, 358], "score": 0.91, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [163, 347, 165, 361], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [70, 358, 542, 444], "lines": [{"bbox": [93, 359, 541, 375], "spans": [{"bbox": [93, 359, 200, 375], "score": 1.0, "content": "Next consider even ", "type": "text"}, {"bbox": [200, 362, 230, 373], "score": 0.93, "content": "r\\neq4", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [230, 359, 281, 375], "score": 1.0, "content": " and odd ", "type": "text"}, {"bbox": [282, 362, 312, 371], "score": 0.9, "content": "k\\,>\\,2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [313, 359, 374, 375], "score": 1.0, "content": ". Either of ", "type": "text"}, {"bbox": [374, 362, 439, 373], "score": 0.92, "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [439, 359, 458, 375], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [459, 360, 527, 373], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [527, 359, 541, 375], "score": 1.0, "content": " is", "type": "text"}], "index": 15}, {"bbox": [69, 373, 542, 392], "spans": [{"bbox": [69, 373, 210, 392], "score": 1.0, "content": "impossible, by comparing ", "type": "text"}, {"bbox": [210, 376, 246, 388], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [247, 373, 275, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [275, 376, 318, 388], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [319, 373, 444, 392], "score": 1.0, "content": " for any simple-current ", "type": "text"}, {"bbox": [444, 376, 453, 385], "score": 0.86, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [453, 373, 542, 392], "score": 1.0, "content": ". For any of the", "type": "text"}], "index": 16}, {"bbox": [70, 387, 541, 405], "spans": [{"bbox": [70, 387, 208, 405], "score": 1.0, "content": "three remaining choices of ", "type": "text"}, {"bbox": [209, 389, 249, 402], "score": 0.93, "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [249, 387, 541, 405], "score": 1.0, "content": ", we can find a simple-current automorphism of the form", "type": "text"}], "index": 17}, {"bbox": [71, 403, 541, 433], "spans": [{"bbox": [71, 403, 123, 433], "score": 0.95, "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "type": "inline_equation", "height": 30, "width": 52}, {"bbox": [124, 409, 252, 425], "score": 1.0, "content": " hitting its inverse onto ", "type": "text"}, {"bbox": [253, 415, 260, 421], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [260, 409, 356, 425], "score": 1.0, "content": " allows us to take ", "type": "text"}, {"bbox": [357, 412, 408, 421], "score": 0.92, "content": "a=b=0", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [408, 409, 452, 425], "score": 1.0, "content": ". Again ", "type": "text"}, {"bbox": [452, 412, 505, 424], "score": 0.95, "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [505, 409, 541, 425], "score": 1.0, "content": " forces", "type": "text"}], "index": 18}, {"bbox": [71, 431, 420, 447], "spans": [{"bbox": [71, 433, 102, 443], "score": 0.87, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [102, 431, 157, 447], "score": 1.0, "content": ", and now ", "type": "text"}, {"bbox": [157, 433, 189, 443], "score": 0.91, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [189, 431, 325, 447], "score": 1.0, "content": " is forbidden. Thus again ", "type": "text"}, {"bbox": [325, 433, 416, 444], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [416, 431, 420, 447], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17}, {"type": "text", "bbox": [70, 444, 541, 488], "lines": [{"bbox": [95, 446, 541, 461], "spans": [{"bbox": [95, 446, 147, 461], "score": 1.0, "content": "As usual, ", "type": "text"}, {"bbox": [147, 448, 176, 457], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [176, 446, 389, 461], "score": 1.0, "content": " is complicated by triality. We can force ", "type": "text"}, {"bbox": [390, 448, 441, 459], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [441, 446, 541, 461], "score": 1.0, "content": ". That we can also", "type": "text"}], "index": 20}, {"bbox": [70, 460, 541, 477], "spans": [{"bbox": [70, 460, 97, 477], "score": 1.0, "content": "take ", "type": "text"}, {"bbox": [97, 461, 154, 473], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [154, 460, 307, 477], "score": 1.0, "content": ", follows from the inequality ", "type": "text"}, {"bbox": [307, 461, 488, 474], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "type": "inline_equation", "height": 13, "width": 181}, {"bbox": [488, 460, 541, 477], "score": 1.0, "content": ", valid for", "type": "text"}], "index": 21}, {"bbox": [71, 473, 442, 492], "spans": [{"bbox": [71, 476, 100, 487], "score": 0.89, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 473, 442, 492], "score": 1.0, "content": ". Establishing that inequality from (3.4) is equivalent to showing", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [98, 500, 514, 516], "lines": [{"bbox": [98, 502, 512, 520], "spans": [{"bbox": [98, 503, 500, 516], "score": 0.77, "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "type": "inline_equation"}, {"bbox": [501, 502, 512, 520], "score": 1.0, "content": "2)", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [69, 528, 398, 542], "lines": [{"bbox": [70, 530, 397, 545], "spans": [{"bbox": [70, 530, 89, 545], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [89, 531, 160, 544], "score": 0.92, "content": "0<x\\le2\\pi/9", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [160, 530, 397, 545], "score": 1.0, "content": ", which can be shown e.g. using Taylor series.", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [70, 543, 541, 657], "lines": [{"bbox": [93, 543, 540, 560], "spans": [{"bbox": [93, 543, 141, 560], "score": 1.0, "content": "For odd ", "type": "text"}, {"bbox": [141, 550, 147, 556], "score": 0.8, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [147, 543, 279, 560], "score": 1.0, "content": ", the charge-conjugation ", "type": "text"}, {"bbox": [279, 547, 289, 556], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [289, 543, 329, 560], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [330, 547, 344, 558], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [344, 543, 508, 560], "score": 1.0, "content": ". Since it must commute with ", "type": "text"}, {"bbox": [509, 550, 516, 556], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 543, 540, 560], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 25}, {"bbox": [69, 556, 541, 574], "spans": [{"bbox": [69, 556, 96, 574], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [96, 558, 198, 573], "score": 0.93, "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 102}, {"bbox": [199, 556, 261, 574], "score": 1.0, "content": " must equal ", "type": "text"}, {"bbox": [262, 559, 352, 573], "score": 0.95, "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [353, 556, 419, 574], "score": 1.0, "content": ", we get that", "type": "text"}, {"bbox": [420, 561, 447, 570], "score": 0.9, "content": "b=0", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [448, 556, 505, 574], "score": 1.0, "content": ". Similarly ", "type": "text"}, {"bbox": [506, 560, 536, 570], "score": 0.92, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [537, 556, 541, 574], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [70, 572, 541, 590], "spans": [{"bbox": [70, 572, 106, 590], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 574, 113, 584], "score": 0.78, "content": "k", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [114, 572, 208, 590], "score": 1.0, "content": " is odd, eliminate ", "type": "text"}, {"bbox": [208, 574, 238, 584], "score": 0.86, "content": "a=1", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [238, 572, 265, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [265, 574, 298, 584], "score": 0.92, "content": "a\\prime=1", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [298, 572, 377, 590], "score": 1.0, "content": " by comparing ", "type": "text"}, {"bbox": [377, 575, 414, 587], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [414, 572, 441, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 575, 485, 587], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [485, 572, 541, 590], "score": 1.0, "content": " as before.", "type": "text"}], "index": 27}, {"bbox": [70, 587, 541, 602], "spans": [{"bbox": [70, 587, 179, 602], "score": 1.0, "content": "The hardest case is ", "type": "text"}, {"bbox": [180, 588, 187, 599], "score": 0.85, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [187, 587, 300, 602], "score": 1.0, "content": " even. We can force ", "type": "text"}, {"bbox": [300, 590, 354, 600], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [354, 587, 445, 602], "score": 1.0, "content": " by hitting with ", "type": "text"}, {"bbox": [445, 589, 466, 601], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [466, 587, 541, 602], "score": 1.0, "content": ". Suppose for", "type": "text"}], "index": 28}, {"bbox": [70, 601, 541, 617], "spans": [{"bbox": [70, 601, 170, 617], "score": 1.0, "content": "contradiction that ", "type": "text"}, {"bbox": [171, 601, 239, 614], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [239, 601, 301, 617], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [302, 603, 376, 616], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [376, 601, 433, 617], "score": 1.0, "content": " (compare ", "type": "text"}, {"bbox": [433, 604, 470, 616], "score": 0.93, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [470, 601, 498, 617], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [498, 603, 536, 616], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [537, 601, 541, 617], "score": 1.0, "content": ",", "type": "text"}], "index": 29}, {"bbox": [69, 615, 540, 633], "spans": [{"bbox": [69, 615, 141, 633], "score": 1.0, "content": "so by (2.7b) ", "type": "text"}, {"bbox": [141, 616, 163, 629], "score": 0.86, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [163, 615, 272, 633], "score": 1.0, "content": " must be a spinor.", "type": "text"}, {"bbox": [273, 617, 390, 630], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [391, 615, 442, 633], "score": 1.0, "content": " requires ", "type": "text"}, {"bbox": [442, 617, 536, 630], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [537, 615, 540, 633], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [69, 630, 541, 646], "spans": [{"bbox": [69, 630, 123, 646], "score": 1.0, "content": "From the ", "type": "text"}, {"bbox": [123, 631, 169, 644], "score": 0.52, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [169, 630, 245, 646], "score": 1.0, "content": " fusion we get ", "type": "text"}, {"bbox": [245, 631, 360, 644], "score": 0.92, "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [360, 630, 388, 646], "score": 1.0, "content": ", but ", "type": "text"}, {"bbox": [389, 633, 438, 641], "score": 0.92, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 8, "width": 49}, {"bbox": [438, 630, 493, 646], "score": 1.0, "content": " says that ", "type": "text"}, {"bbox": [493, 632, 541, 644], "score": 0.9, "content": "\\pi\\Lambda_{r-1}=", "type": "inline_equation", "height": 12, "width": 48}], "index": 31}, {"bbox": [71, 644, 251, 662], "spans": [{"bbox": [71, 644, 147, 659], "score": 0.91, "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 15, "width": 76}, {"bbox": [148, 644, 251, 662], "score": 1.0, "content": " \u2014 a contradiction.", "type": "text"}], "index": 32}], "index": 28.5}, {"type": "text", "bbox": [70, 658, 541, 715], "lines": [{"bbox": [93, 658, 541, 675], "spans": [{"bbox": [93, 658, 239, 675], "score": 1.0, "content": "Thus in all cases we have ", "type": "text"}, {"bbox": [240, 660, 340, 672], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [341, 658, 407, 675], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [408, 660, 486, 673], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [486, 658, 541, 675], "score": 1.0, "content": " (compare", "type": "text"}], "index": 33}, {"bbox": [71, 673, 541, 689], "spans": [{"bbox": [71, 673, 109, 687], "score": 0.91, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [109, 674, 137, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [137, 673, 175, 687], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [175, 674, 198, 689], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [198, 675, 219, 686], "score": 0.9, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [220, 674, 410, 689], "score": 1.0, "content": " is a spinor and in fact must equal ", "type": "text"}, {"bbox": [411, 674, 491, 687], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [491, 674, 541, 689], "score": 1.0, "content": ". Hitting", "type": "text"}], "index": 34}, {"bbox": [70, 687, 542, 704], "spans": [{"bbox": [70, 687, 98, 704], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 687, 147, 702], "score": 0.93, "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "type": "inline_equation", "height": 15, "width": 48}, {"bbox": [147, 687, 234, 704], "score": 1.0, "content": ", we can require ", "type": "text"}, {"bbox": [234, 690, 285, 700], "score": 0.91, "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [285, 687, 322, 704], "score": 1.0, "content": ". That ", "type": "text"}, {"bbox": [323, 690, 356, 701], "score": 0.92, "content": "\\pi\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [356, 687, 445, 704], "score": 1.0, "content": " must now equal ", "type": "text"}, {"bbox": [446, 690, 471, 701], "score": 0.92, "content": "\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [472, 687, 542, 704], "score": 1.0, "content": " follows from", "type": "text"}], "index": 35}, {"bbox": [71, 703, 177, 717], "spans": [{"bbox": [71, 703, 91, 717], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [92, 703, 137, 715], "score": 0.28, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [138, 703, 177, 717], "score": 1.0, "content": " fusion.", "type": "text"}], "index": 36}], "index": 34.5}], "layout_bboxes": [], "page_idx": 18, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 741], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "19", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 70, 542, 100], "lines": [], "index": 0.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [71, 72, 542, 102], "lines_deleted": true}, {"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_18", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 149, 541, 178], "lines": [{"bbox": [70, 151, 539, 168], "spans": [{"bbox": [70, 151, 100, 168], "score": 1.0, "content": "as in ", "type": "text"}, {"bbox": [100, 153, 121, 165], "score": 0.41, "content": "\\S4.3", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [122, 151, 165, 168], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [165, 154, 195, 163], "score": 0.92, "content": "r=k", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [195, 151, 360, 168], "score": 1.0, "content": ", that inequality only holds for ", "type": "text"}, {"bbox": [360, 153, 389, 164], "score": 0.88, "content": "\\ell>1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [390, 151, 487, 168], "score": 1.0, "content": ", but we can force ", "type": "text"}, {"bbox": [488, 154, 539, 165], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 51}], "index": 3}, {"bbox": [70, 166, 254, 183], "spans": [{"bbox": [70, 166, 127, 183], "score": 1.0, "content": "by hitting ", "type": "text"}, {"bbox": [127, 172, 135, 178], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [135, 166, 229, 183], "score": 1.0, "content": " if necessary with ", "type": "text"}, {"bbox": [229, 172, 249, 179], "score": 0.88, "content": "\\pi_{\\mathrm{rld}}", "type": "inline_equation", "height": 7, "width": 20}, {"bbox": [249, 166, 254, 183], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [70, 151, 539, 183]}, {"type": "text", "bbox": [70, 179, 541, 208], "lines": [{"bbox": [93, 180, 541, 197], "spans": [{"bbox": [93, 180, 203, 197], "score": 1.0, "content": "The remaining case ", "type": "text"}, {"bbox": [203, 182, 225, 195], "score": 0.91, "content": "C_{2,3}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [225, 180, 314, 197], "score": 1.0, "content": " follows because ", "type": "text"}, {"bbox": [314, 182, 371, 192], "score": 0.92, "content": "\\pi^{\\prime}J0\\,=\\,J0", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [372, 180, 434, 197], "score": 1.0, "content": ": by (2.7b) ", "type": "text"}, {"bbox": [434, 182, 495, 194], "score": 0.87, "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [495, 180, 541, 197], "score": 1.0, "content": ", and by", "type": "text"}], "index": 5}, {"bbox": [72, 195, 292, 210], "spans": [{"bbox": [72, 196, 105, 210], "score": 1.0, "content": "(2.7a) ", "type": "text"}, {"bbox": [105, 195, 162, 208], "score": 0.89, "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [163, 196, 170, 210], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [171, 195, 191, 208], "score": 0.88, "content": "3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [191, 196, 216, 210], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [216, 196, 225, 206], "score": 0.84, "content": "J", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [225, 196, 292, 210], "score": 1.0, "content": "-fixed-point).", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [72, 180, 541, 210]}, {"type": "title", "bbox": [71, 221, 219, 236], "lines": [{"bbox": [71, 225, 219, 236], "spans": [{"bbox": [71, 225, 119, 236], "score": 1.0, "content": "4.5. The ", "type": "text"}, {"bbox": [119, 225, 130, 235], "score": 0.8, "content": "D", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [131, 225, 219, 236], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_18", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 243, 541, 300], "lines": [{"bbox": [95, 244, 541, 259], "spans": [{"bbox": [95, 247, 124, 256], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 244, 201, 259], "score": 1.0, "content": " is trivial, and ", "type": "text"}, {"bbox": [202, 245, 231, 256], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [232, 244, 398, 259], "score": 1.0, "content": " will be considered shortly. For ", "type": "text"}, {"bbox": [398, 247, 427, 256], "score": 0.9, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [427, 244, 541, 259], "score": 1.0, "content": ", Proposition 4.1 tells", "type": "text"}], "index": 8}, {"bbox": [70, 258, 541, 274], "spans": [{"bbox": [70, 258, 111, 274], "score": 1.0, "content": "us that", "type": "text"}, {"bbox": [112, 259, 189, 273], "score": 0.93, "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 77}, {"bbox": [190, 258, 216, 274], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [216, 258, 303, 273], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [303, 258, 329, 274], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [329, 260, 419, 273], "score": 0.93, "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [419, 258, 541, 274], "score": 1.0, "content": ". Immediate from (3.4)", "type": "text"}], "index": 9}, {"bbox": [69, 272, 542, 289], "spans": [{"bbox": [69, 272, 109, 289], "score": 1.0, "content": "is that ", "type": "text"}, {"bbox": [110, 274, 173, 288], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [173, 272, 226, 289], "score": 1.0, "content": " and that ", "type": "text"}, {"bbox": [226, 275, 261, 287], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\psi]", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [262, 272, 334, 289], "score": 1.0, "content": ", for a spinor ", "type": "text"}, {"bbox": [334, 275, 343, 287], "score": 0.89, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [343, 272, 469, 289], "score": 1.0, "content": ", takes its maximum at ", "type": "text"}, {"bbox": [469, 274, 509, 287], "score": 0.93, "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [509, 272, 542, 289], "score": 1.0, "content": ". Our", "type": "text"}], "index": 10}, {"bbox": [70, 288, 502, 303], "spans": [{"bbox": [70, 288, 176, 303], "score": 1.0, "content": "first step is to force ", "type": "text"}, {"bbox": [177, 289, 268, 301], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [268, 288, 502, 303], "score": 1.0, "content": ". Unfortunately this requires a case analysis.", "type": "text"}], "index": 11}], "index": 9.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [69, 244, 542, 303]}, {"type": "text", "bbox": [69, 301, 541, 358], "lines": [{"bbox": [94, 301, 542, 319], "spans": [{"bbox": [94, 301, 201, 319], "score": 1.0, "content": "Consider first even ", "type": "text"}, {"bbox": [201, 304, 235, 315], "score": 0.92, "content": "r\\,\\neq\\,4", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [235, 301, 297, 319], "score": 1.0, "content": ", and even ", "type": "text"}, {"bbox": [297, 304, 331, 313], "score": 0.88, "content": "k\\ >\\ 2", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [332, 301, 376, 319], "score": 1.0, "content": ". Now, ", "type": "text"}, {"bbox": [377, 304, 504, 316], "score": 0.92, "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 127}, {"bbox": [504, 301, 542, 319], "score": 1.0, "content": " forces", "type": "text"}], "index": 12}, {"bbox": [71, 317, 541, 347], "spans": [{"bbox": [71, 325, 104, 335], "score": 0.9, "content": "b=b^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [104, 324, 394, 339], "score": 1.0, "content": "; hence hitting with the simple-current automorphism ", "type": "text"}, {"bbox": [394, 317, 449, 347], "score": 0.94, "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [450, 326, 541, 338], "score": 1.0, "content": ", we may assume", "type": "text"}], "index": 13}, {"bbox": [71, 347, 165, 361], "spans": [{"bbox": [71, 347, 162, 358], "score": 0.91, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [163, 347, 165, 361], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [71, 301, 542, 361]}, {"type": "text", "bbox": [70, 358, 542, 444], "lines": [{"bbox": [93, 359, 541, 375], "spans": [{"bbox": [93, 359, 200, 375], "score": 1.0, "content": "Next consider even ", "type": "text"}, {"bbox": [200, 362, 230, 373], "score": 0.93, "content": "r\\neq4", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [230, 359, 281, 375], "score": 1.0, "content": " and odd ", "type": "text"}, {"bbox": [282, 362, 312, 371], "score": 0.9, "content": "k\\,>\\,2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [313, 359, 374, 375], "score": 1.0, "content": ". Either of ", "type": "text"}, {"bbox": [374, 362, 439, 373], "score": 0.92, "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [439, 359, 458, 375], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [459, 360, 527, 373], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [527, 359, 541, 375], "score": 1.0, "content": " is", "type": "text"}], "index": 15}, {"bbox": [69, 373, 542, 392], "spans": [{"bbox": [69, 373, 210, 392], "score": 1.0, "content": "impossible, by comparing ", "type": "text"}, {"bbox": [210, 376, 246, 388], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [247, 373, 275, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [275, 376, 318, 388], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [319, 373, 444, 392], "score": 1.0, "content": " for any simple-current ", "type": "text"}, {"bbox": [444, 376, 453, 385], "score": 0.86, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [453, 373, 542, 392], "score": 1.0, "content": ". For any of the", "type": "text"}], "index": 16}, {"bbox": [70, 387, 541, 405], "spans": [{"bbox": [70, 387, 208, 405], "score": 1.0, "content": "three remaining choices of ", "type": "text"}, {"bbox": [209, 389, 249, 402], "score": 0.93, "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [249, 387, 541, 405], "score": 1.0, "content": ", we can find a simple-current automorphism of the form", "type": "text"}], "index": 17}, {"bbox": [71, 403, 541, 433], "spans": [{"bbox": [71, 403, 123, 433], "score": 0.95, "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "type": "inline_equation", "height": 30, "width": 52}, {"bbox": [124, 409, 252, 425], "score": 1.0, "content": " hitting its inverse onto ", "type": "text"}, {"bbox": [253, 415, 260, 421], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [260, 409, 356, 425], "score": 1.0, "content": " allows us to take ", "type": "text"}, {"bbox": [357, 412, 408, 421], "score": 0.92, "content": "a=b=0", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [408, 409, 452, 425], "score": 1.0, "content": ". Again ", "type": "text"}, {"bbox": [452, 412, 505, 424], "score": 0.95, "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [505, 409, 541, 425], "score": 1.0, "content": " forces", "type": "text"}], "index": 18}, {"bbox": [71, 431, 420, 447], "spans": [{"bbox": [71, 433, 102, 443], "score": 0.87, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [102, 431, 157, 447], "score": 1.0, "content": ", and now ", "type": "text"}, {"bbox": [157, 433, 189, 443], "score": 0.91, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [189, 431, 325, 447], "score": 1.0, "content": " is forbidden. Thus again ", "type": "text"}, {"bbox": [325, 433, 416, 444], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [416, 431, 420, 447], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [69, 359, 542, 447]}, {"type": "text", "bbox": [70, 444, 541, 488], "lines": [{"bbox": [95, 446, 541, 461], "spans": [{"bbox": [95, 446, 147, 461], "score": 1.0, "content": "As usual, ", "type": "text"}, {"bbox": [147, 448, 176, 457], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [176, 446, 389, 461], "score": 1.0, "content": " is complicated by triality. We can force ", "type": "text"}, {"bbox": [390, 448, 441, 459], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [441, 446, 541, 461], "score": 1.0, "content": ". That we can also", "type": "text"}], "index": 20}, {"bbox": [70, 460, 541, 477], "spans": [{"bbox": [70, 460, 97, 477], "score": 1.0, "content": "take ", "type": "text"}, {"bbox": [97, 461, 154, 473], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [154, 460, 307, 477], "score": 1.0, "content": ", follows from the inequality ", "type": "text"}, {"bbox": [307, 461, 488, 474], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "type": "inline_equation", "height": 13, "width": 181}, {"bbox": [488, 460, 541, 477], "score": 1.0, "content": ", valid for", "type": "text"}], "index": 21}, {"bbox": [71, 473, 442, 492], "spans": [{"bbox": [71, 476, 100, 487], "score": 0.89, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 473, 442, 492], "score": 1.0, "content": ". Establishing that inequality from (3.4) is equivalent to showing", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [70, 446, 541, 492]}, {"type": "text", "bbox": [98, 500, 514, 516], "lines": [{"bbox": [98, 502, 512, 520], "spans": [{"bbox": [98, 503, 500, 516], "score": 0.77, "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "type": "inline_equation"}, {"bbox": [501, 502, 512, 520], "score": 1.0, "content": "2)", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [98, 502, 512, 520]}, {"type": "text", "bbox": [69, 528, 398, 542], "lines": [{"bbox": [70, 530, 397, 545], "spans": [{"bbox": [70, 530, 89, 545], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [89, 531, 160, 544], "score": 0.92, "content": "0<x\\le2\\pi/9", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [160, 530, 397, 545], "score": 1.0, "content": ", which can be shown e.g. using Taylor series.", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [70, 530, 397, 545]}, {"type": "text", "bbox": [70, 543, 541, 657], "lines": [{"bbox": [93, 543, 540, 560], "spans": [{"bbox": [93, 543, 141, 560], "score": 1.0, "content": "For odd ", "type": "text"}, {"bbox": [141, 550, 147, 556], "score": 0.8, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [147, 543, 279, 560], "score": 1.0, "content": ", the charge-conjugation ", "type": "text"}, {"bbox": [279, 547, 289, 556], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [289, 543, 329, 560], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [330, 547, 344, 558], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [344, 543, 508, 560], "score": 1.0, "content": ". Since it must commute with ", "type": "text"}, {"bbox": [509, 550, 516, 556], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 543, 540, 560], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 25}, {"bbox": [69, 556, 541, 574], "spans": [{"bbox": [69, 556, 96, 574], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [96, 558, 198, 573], "score": 0.93, "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 102}, {"bbox": [199, 556, 261, 574], "score": 1.0, "content": " must equal ", "type": "text"}, {"bbox": [262, 559, 352, 573], "score": 0.95, "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [353, 556, 419, 574], "score": 1.0, "content": ", we get that", "type": "text"}, {"bbox": [420, 561, 447, 570], "score": 0.9, "content": "b=0", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [448, 556, 505, 574], "score": 1.0, "content": ". Similarly ", "type": "text"}, {"bbox": [506, 560, 536, 570], "score": 0.92, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [537, 556, 541, 574], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [70, 572, 541, 590], "spans": [{"bbox": [70, 572, 106, 590], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 574, 113, 584], "score": 0.78, "content": "k", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [114, 572, 208, 590], "score": 1.0, "content": " is odd, eliminate ", "type": "text"}, {"bbox": [208, 574, 238, 584], "score": 0.86, "content": "a=1", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [238, 572, 265, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [265, 574, 298, 584], "score": 0.92, "content": "a\\prime=1", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [298, 572, 377, 590], "score": 1.0, "content": " by comparing ", "type": "text"}, {"bbox": [377, 575, 414, 587], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [414, 572, 441, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 575, 485, 587], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [485, 572, 541, 590], "score": 1.0, "content": " as before.", "type": "text"}], "index": 27}, {"bbox": [70, 587, 541, 602], "spans": [{"bbox": [70, 587, 179, 602], "score": 1.0, "content": "The hardest case is ", "type": "text"}, {"bbox": [180, 588, 187, 599], "score": 0.85, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [187, 587, 300, 602], "score": 1.0, "content": " even. We can force ", "type": "text"}, {"bbox": [300, 590, 354, 600], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [354, 587, 445, 602], "score": 1.0, "content": " by hitting with ", "type": "text"}, {"bbox": [445, 589, 466, 601], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [466, 587, 541, 602], "score": 1.0, "content": ". Suppose for", "type": "text"}], "index": 28}, {"bbox": [70, 601, 541, 617], "spans": [{"bbox": [70, 601, 170, 617], "score": 1.0, "content": "contradiction that ", "type": "text"}, {"bbox": [171, 601, 239, 614], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [239, 601, 301, 617], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [302, 603, 376, 616], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [376, 601, 433, 617], "score": 1.0, "content": " (compare ", "type": "text"}, {"bbox": [433, 604, 470, 616], "score": 0.93, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [470, 601, 498, 617], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [498, 603, 536, 616], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [537, 601, 541, 617], "score": 1.0, "content": ",", "type": "text"}], "index": 29}, {"bbox": [69, 615, 540, 633], "spans": [{"bbox": [69, 615, 141, 633], "score": 1.0, "content": "so by (2.7b) ", "type": "text"}, {"bbox": [141, 616, 163, 629], "score": 0.86, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [163, 615, 272, 633], "score": 1.0, "content": " must be a spinor.", "type": "text"}, {"bbox": [273, 617, 390, 630], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [391, 615, 442, 633], "score": 1.0, "content": " requires ", "type": "text"}, {"bbox": [442, 617, 536, 630], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [537, 615, 540, 633], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [69, 630, 541, 646], "spans": [{"bbox": [69, 630, 123, 646], "score": 1.0, "content": "From the ", "type": "text"}, {"bbox": [123, 631, 169, 644], "score": 0.52, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [169, 630, 245, 646], "score": 1.0, "content": " fusion we get ", "type": "text"}, {"bbox": [245, 631, 360, 644], "score": 0.92, "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [360, 630, 388, 646], "score": 1.0, "content": ", but ", "type": "text"}, {"bbox": [389, 633, 438, 641], "score": 0.92, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 8, "width": 49}, {"bbox": [438, 630, 493, 646], "score": 1.0, "content": " says that ", "type": "text"}, {"bbox": [493, 632, 541, 644], "score": 0.9, "content": "\\pi\\Lambda_{r-1}=", "type": "inline_equation", "height": 12, "width": 48}], "index": 31}, {"bbox": [71, 644, 251, 662], "spans": [{"bbox": [71, 644, 147, 659], "score": 0.91, "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 15, "width": 76}, {"bbox": [148, 644, 251, 662], "score": 1.0, "content": " \u2014 a contradiction.", "type": "text"}], "index": 32}], "index": 28.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [69, 543, 541, 662]}, {"type": "text", "bbox": [70, 658, 541, 715], "lines": [{"bbox": [93, 658, 541, 675], "spans": [{"bbox": [93, 658, 239, 675], "score": 1.0, "content": "Thus in all cases we have ", "type": "text"}, {"bbox": [240, 660, 340, 672], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [341, 658, 407, 675], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [408, 660, 486, 673], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [486, 658, 541, 675], "score": 1.0, "content": " (compare", "type": "text"}], "index": 33}, {"bbox": [71, 673, 541, 689], "spans": [{"bbox": [71, 673, 109, 687], "score": 0.91, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [109, 674, 137, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [137, 673, 175, 687], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [175, 674, 198, 689], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [198, 675, 219, 686], "score": 0.9, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [220, 674, 410, 689], "score": 1.0, "content": " is a spinor and in fact must equal ", "type": "text"}, {"bbox": [411, 674, 491, 687], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [491, 674, 541, 689], "score": 1.0, "content": ". Hitting", "type": "text"}], "index": 34}, {"bbox": [70, 687, 542, 704], "spans": [{"bbox": [70, 687, 98, 704], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 687, 147, 702], "score": 0.93, "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "type": "inline_equation", "height": 15, "width": 48}, {"bbox": [147, 687, 234, 704], "score": 1.0, "content": ", we can require ", "type": "text"}, {"bbox": [234, 690, 285, 700], "score": 0.91, "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [285, 687, 322, 704], "score": 1.0, "content": ". That ", "type": "text"}, {"bbox": [323, 690, 356, 701], "score": 0.92, "content": "\\pi\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [356, 687, 445, 704], "score": 1.0, "content": " must now equal ", "type": "text"}, {"bbox": [446, 690, 471, 701], "score": 0.92, "content": "\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [472, 687, 542, 704], "score": 1.0, "content": " follows from", "type": "text"}], "index": 35}, {"bbox": [71, 703, 177, 717], "spans": [{"bbox": [71, 703, 91, 717], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [92, 703, 137, 715], "score": 0.28, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [138, 703, 177, 717], "score": 1.0, "content": " fusion.", "type": "text"}], "index": 36}], "index": 34.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [70, 658, 542, 717]}]}
0002044v1	21	where $$[x]$$ here denotes the greatest integer not more than $$x$$ . The absolute value of each of these is quickly seen to be greater than 1 unless $$\ell\,\equiv\,\pm1$$ (mod $$2\kappa$$ ), except for the orthogonal algebras when $$k\ \leq\ 2$$ . An isomorphism $$\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(X_{r^{\prime},k^{\prime}})$$ would require then that whenever $$\ell\equiv\pm1$$ (mod $$2\kappa$$ ) is coprime to $$\kappa^{\prime}$$ , it must also satisfy $$\ell\equiv\pm1$$ (mod $$2\kappa^{\prime}$$ ), and conversely. This forces $$\kappa=\kappa^{\prime}$$ , for $$X=B$$ or $$D$$ and $$k>2$$ , or $$X=C$$ and any $$k$$ . If $$\mathscr{R}(C_{r,k})\cong\mathscr{R}(C_{s,m})$$ , then that Galois argument implies $$r+k+1=s+m+1$$ , so compare numbers of highest-weights: $$\big(\mathbf{\Lambda}_{r}^{r+k}\big)=\big(\mathbf{\Lambda}_{s}^{r+k}\big)$$ . A similar argument works for the orthogonal algebras. For instance suppose $$\mathcal{R}(B_{r,k})\cong$$ $$\mathcal{R}(B_{s,m})$$ but $$B_{r,k}\ne B_{s,m}$$ , and that $$k,m\,>\,2$$ . Then Galois implies $$2r+k\,=\,2s\,+\,m$$ . Comparing the value of $$\mathcal{D}(\Lambda_{1})$$ (the second smallest q-dimension when $$k>3$$ ), using (3.2) with $$\lambda=0$$ , tells us that $$2s+1=k,2r+1=m$$ . Now count the number of fixed-points of $$J$$ in both cases: $$\binom{\kappa/2-1}{r-1}=\binom{\kappa/2-1}{s-1}$$ , i.e. $$s-1=(k-1)/2$$ , a contradiction. For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $$\Lambda_{\star}$$ × $$\Lambda_{\star}$$ (when $$\Lambda_{\star}$$ has second smallest q-dimension). For example, for $$A_{1,k}$$ $$\left(k>1\right)$$ , $$C_{r,k}$$ ( $$k>1$$ , except for $$C_{2,2},C_{2,3},C_{3,2})$$ , and $$E_{7,k}$$ ( $$k>4)$$ ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic. For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $$r$$ and $$r-1$$ for $$B_{r,2}$$ and $$D_{r,2}$$ , except for $$D_{4,2}$$ ). For the exceptional algebras, comparing $$\mathcal{D}(\boldsymbol{\Lambda}_{\star})$$ and the number of highest-weights is effective. Recall that both $$\|\|P_{+}\|\|$$ and $$\mathcal{D}(\boldsymbol{\Lambda}_{\star})$$ for a fixed algebra monotonically increase with $$k$$ to (respectively) $$\infty$$ and the Weyl dimension of $$\Lambda_{\star}$$ , which is 7, 26, and 248 for $$G_{2},F_{4},E_{8}$$ respectively. For $$E_{8,k}$$ , $$\mathcal{D}(\Lambda_{1})$$ exceeds 7 for $$k\geq5$$ , and exceeds 26 for $$k\geq11$$ , while $$F_{4,k}$$ exceeds 7 for $$k\geq4$$ . The number of highest-weights of $$E_{8,4},E_{8,10}$$ , and $$F_{4,3}$$ are 10, 135, and 9, so only a small number of possibilities need be considered. # References 1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas- sification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998), 163–166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99–136. 2. D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B288 (1987), 628–648. 3. J. B¨ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999. 4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B323 (1994), 316–321. 5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hep- th/0001158, 2000. 6. Ph. Di Francesco, P. Mathieu and D. S´en´echal, “Conformal Field Theory”, Springer-Verlag, New York, 1997. 7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374.	<p>where $$[x]$$ here denotes the greatest integer not more than $$x$$ . The absolute value of each of these is quickly seen to be greater than 1 unless $$\ell\,\equiv\,\pm1$$ (mod $$2\kappa$$ ), except for the orthogonal algebras when $$k\ \leq\ 2$$ . An isomorphism $$\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(X_{r^{\prime},k^{\prime}})$$ would require then that whenever $$\ell\equiv\pm1$$ (mod $$2\kappa$$ ) is coprime to $$\kappa^{\prime}$$ , it must also satisfy $$\ell\equiv\pm1$$ (mod $$2\kappa^{\prime}$$ ), and conversely. This forces $$\kappa=\kappa^{\prime}$$ , for $$X=B$$ or $$D$$ and $$k>2$$ , or $$X=C$$ and any $$k$$ .</p> <p>If $$\mathscr{R}(C_{r,k})\cong\mathscr{R}(C_{s,m})$$ , then that Galois argument implies $$r+k+1=s+m+1$$ , so compare numbers of highest-weights: $$\big(\mathbf{\Lambda}_{r}^{r+k}\big)=\big(\mathbf{\Lambda}_{s}^{r+k}\big)$$ .</p> <p>A similar argument works for the orthogonal algebras. For instance suppose $$\mathcal{R}(B_{r,k})\cong$$ $$\mathcal{R}(B_{s,m})$$ but $$B_{r,k}\ne B_{s,m}$$ , and that $$k,m\,>\,2$$ . Then Galois implies $$2r+k\,=\,2s\,+\,m$$ . Comparing the value of $$\mathcal{D}(\Lambda_{1})$$ (the second smallest q-dimension when $$k>3$$ ), using (3.2) with $$\lambda=0$$ , tells us that $$2s+1=k,2r+1=m$$ . Now count the number of fixed-points of $$J$$ in both cases: $$\binom{\kappa/2-1}{r-1}=\binom{\kappa/2-1}{s-1}$$ , i.e. $$s-1=(k-1)/2$$ , a contradiction.</p> <p>For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $$\Lambda_{\star}$$ × $$\Lambda_{\star}$$ (when $$\Lambda_{\star}$$ has second smallest q-dimension). For example, for $$A_{1,k}$$ $$\left(k>1\right)$$ , $$C_{r,k}$$ ( $$k>1$$ , except for $$C_{2,2},C_{2,3},C_{3,2})$$ , and $$E_{7,k}$$ ( $$k>4)$$ ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic.</p> <p>For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $$r$$ and $$r-1$$ for $$B_{r,2}$$ and $$D_{r,2}$$ , except for $$D_{4,2}$$ ).</p> <p>For the exceptional algebras, comparing $$\mathcal{D}(\boldsymbol{\Lambda}_{\star})$$ and the number of highest-weights is effective. Recall that both $$\|\|P_{+}\|\|$$ and $$\mathcal{D}(\boldsymbol{\Lambda}_{\star})$$ for a fixed algebra monotonically increase with $$k$$ to (respectively) $$\infty$$ and the Weyl dimension of $$\Lambda_{\star}$$ , which is 7, 26, and 248 for $$G_{2},F_{4},E_{8}$$ respectively. For $$E_{8,k}$$ , $$\mathcal{D}(\Lambda_{1})$$ exceeds 7 for $$k\geq5$$ , and exceeds 26 for $$k\geq11$$ , while $$F_{4,k}$$ exceeds 7 for $$k\geq4$$ . The number of highest-weights of $$E_{8,4},E_{8,10}$$ , and $$F_{4,3}$$ are 10, 135, and 9, so only a small number of possibilities need be considered.</p> <h1>References</h1> <p>1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas- sification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998), 163–166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99–136. 2. D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B288 (1987), 628–648. 3. J. B¨ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999. 4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B323 (1994), 316–321. 5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hep- th/0001158, 2000. 6. Ph. Di Francesco, P. Mathieu and D. S´en´echal, “Conformal Field Theory”, Springer-Verlag, New York, 1997. 7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374.</p>	[{"type": "text", "coordinates": [71, 70, 541, 143], "content": "where $$[x]$$ here denotes the greatest integer not more than $$x$$ . The absolute value of each\nof these is quickly seen to be greater than 1 unless $$\\ell\\,\\equiv\\,\\pm1$$ (mod $$2\\kappa$$ ), except for the\northogonal algebras when $$k\\ \\leq\\ 2$$ . An isomorphism $$\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})$$ would require\nthen that whenever $$\\ell\\equiv\\pm1$$ (mod $$2\\kappa$$ ) is coprime to $$\\kappa^{\\prime}$$ , it must also satisfy $$\\ell\\equiv\\pm1$$ (mod\n$$2\\kappa^{\\prime}$$ ), and conversely. This forces $$\\kappa=\\kappa^{\\prime}$$ , for $$X=B$$ or $$D$$ and $$k>2$$ , or $$X=C$$ and any $$k$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 144, 541, 175], "content": "If $$\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})$$ , then that Galois argument implies $$r+k+1=s+m+1$$ , so\ncompare numbers of highest-weights: $$\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 176, 541, 248], "content": "A similar argument works for the orthogonal algebras. For instance suppose $$\\mathcal{R}(B_{r,k})\\cong$$\n$$\\mathcal{R}(B_{s,m})$$ but $$B_{r,k}\\ne B_{s,m}$$ , and that $$k,m\\,>\\,2$$ . Then Galois implies $$2r+k\\,=\\,2s\\,+\\,m$$ .\nComparing the value of $$\\mathcal{D}(\\Lambda_{1})$$ (the second smallest q-dimension when $$k>3$$ ), using (3.2)\nwith $$\\lambda=0$$ , tells us that $$2s+1=k,2r+1=m$$ . Now count the number of fixed-points of\n$$J$$ in both cases: $$\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}$$ , i.e. $$s-1=(k-1)/2$$ , a contradiction.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [71, 249, 541, 319], "content": "For comparing classical algebras with exceptional algebras, a useful device is to count\nthe number of weights appearing in the fusion $$\\Lambda_{\\star}$$ \u00d7 $$\\Lambda_{\\star}$$ (when $$\\Lambda_{\\star}$$ has second smallest\nq-dimension). For example, for $$A_{1,k}$$ $$\\left(k>1\\right)$$ , $$C_{r,k}$$ ( $$k>1$$ , except for $$C_{2,2},C_{2,3},C_{3,2})$$ , and\n$$E_{7,k}$$ ( $$k>4)$$ ), we learned in \u00a73 that this number is 2, 3, 4 respectively, so none of these can\nbe isomorphic.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 321, 541, 349], "content": "For the orthogonal algebras at level 2, useful is the number of weights with second\nsmallest q-dimension (respectively $$r$$ and $$r-1$$ for $$B_{r,2}$$ and $$D_{r,2}$$ , except for $$D_{4,2}$$ ).", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [70, 351, 541, 436], "content": "For the exceptional algebras, comparing $$\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$$ and the number of highest-weights is\neffective. Recall that both $$\|\|P_{+}\|\|$$ and $$\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$$ for a fixed algebra monotonically increase with\n$$k$$ to (respectively) $$\\infty$$ and the Weyl dimension of $$\\Lambda_{\\star}$$ , which is 7, 26, and 248 for $$G_{2},F_{4},E_{8}$$\nrespectively. For $$E_{8,k}$$ , $$\\mathcal{D}(\\Lambda_{1})$$ exceeds 7 for $$k\\geq5$$ , and exceeds 26 for $$k\\geq11$$ , while $$F_{4,k}$$\nexceeds 7 for $$k\\geq4$$ . The number of highest-weights of $$E_{8,4},E_{8,10}$$ , and $$F_{4,3}$$ are 10, 135,\nand 9, so only a small number of possibilities need be considered.", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [270, 455, 342, 468], "content": "References", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [78, 478, 541, 716], "content": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas-\nsification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998),\n163\u2013166;\nJ. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds,\nNucl. Phys. B530 (1998), 99\u2013136.\n2. D. Bernard, String characters from Kac\u2013Moody automorphisms, Nucl. Phys. B288\n(1987), 628\u2013648.\n3. J. B\u00a8ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I\ncoupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999.\n4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories,\nPhys. Lett. B323 (1994), 316\u2013321.\n5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hep-\nth/0001158, 2000.\n6. Ph. Di Francesco, P. Mathieu and D. S\u00b4en\u00b4echal, \u201cConformal Field Theory\u201d,\nSpringer-Verlag, New York, 1997.\n7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347\u2013374.", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [71, 73, 106, 89], "content": "where ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [106, 75, 120, 87], "content": "[x]", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [120, 73, 381, 89], "content": " here denotes the greatest integer not more than ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [381, 78, 388, 84], "content": "x", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [389, 73, 541, 89], "content": ". The absolute value of each", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 88, 356, 104], "content": "of these is quickly seen to be greater than 1 unless ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [357, 90, 398, 100], "content": "\\ell\\,\\equiv\\,\\pm1", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [399, 88, 437, 104], "content": " (mod ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [437, 90, 450, 99], "content": "2\\kappa", "score": 0.72, "index": 9}, {"type": "text", "coordinates": [451, 88, 541, 104], "content": "), except for the", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [71, 102, 212, 118], "content": "orthogonal algebras when ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [213, 104, 246, 115], "content": "k\\ \\leq\\ 2", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [246, 102, 349, 118], "content": ". An isomorphism ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [350, 104, 462, 116], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [462, 102, 541, 118], "content": " would require", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [71, 117, 177, 131], "content": "then that whenever ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [178, 118, 216, 128], "content": "\\ell\\equiv\\pm1", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [216, 117, 252, 131], "content": " (mod ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [252, 119, 266, 127], "content": "2\\kappa", "score": 0.66, "index": 19}, {"type": "text", "coordinates": [266, 117, 347, 131], "content": ") is coprime to ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [347, 118, 357, 127], "content": "\\kappa^{\\prime}", "score": 0.89, "index": 21}, {"type": "text", "coordinates": [358, 117, 469, 131], "content": ", it must also satisfy ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [470, 118, 507, 128], "content": "\\ell\\equiv\\pm1", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [508, 117, 540, 131], "content": " (mod", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [71, 132, 88, 142], "content": "2\\kappa^{\\prime}", "score": 0.77, "index": 25}, {"type": "text", "coordinates": [88, 131, 243, 145], "content": "), and conversely. This forces ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [244, 132, 277, 142], "content": "\\kappa=\\kappa^{\\prime}", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [277, 131, 302, 145], "content": ", for ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [302, 133, 339, 142], "content": "X=B", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [339, 131, 356, 145], "content": " or ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [357, 133, 367, 142], "content": "D", "score": 0.91, "index": 31}, {"type": "text", "coordinates": [367, 131, 393, 145], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [394, 133, 423, 142], "content": "k>2", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [423, 131, 443, 145], "content": ", or ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [444, 133, 480, 142], "content": "X=C", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [480, 131, 529, 145], "content": " and any ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [529, 133, 536, 142], "content": "k", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [537, 131, 540, 145], "content": ".", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [94, 146, 107, 163], "content": "If ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [107, 147, 210, 160], "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [210, 146, 403, 163], "content": ", then that Galois argument implies ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [403, 148, 521, 158], "content": "r+k+1=s+m+1", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [522, 146, 542, 163], "content": ", so", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [70, 160, 269, 179], "content": "compare numbers of highest-weights: ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [270, 161, 345, 177], "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "score": 0.96, "index": 45}, {"type": "text", "coordinates": [345, 160, 349, 179], "content": ".", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [94, 176, 485, 193], "content": "A similar argument works for the orthogonal algebras. For instance suppose ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [485, 178, 542, 191], "content": "\\mathcal{R}(B_{r,k})\\cong", "score": 0.93, "index": 48}, {"type": "inline_equation", "coordinates": [71, 193, 117, 205], "content": "\\mathcal{R}(B_{s,m})", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [117, 191, 144, 207], "content": " but ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [144, 194, 212, 205], "content": "B_{r,k}\\ne B_{s,m}", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [212, 191, 271, 207], "content": ", and that ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [271, 193, 320, 205], "content": "k,m\\,>\\,2", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [321, 191, 443, 207], "content": ". Then Galois implies ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [444, 194, 536, 204], "content": "2r+k\\,=\\,2s\\,+\\,m", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [536, 191, 541, 207], "content": ".", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [72, 207, 198, 221], "content": "Comparing the value of ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [198, 207, 231, 220], "content": "\\mathcal{D}(\\Lambda_{1})", "score": 0.94, "index": 58}, {"type": "text", "coordinates": [231, 207, 442, 221], "content": " (the second smallest q-dimension when ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [443, 208, 472, 217], "content": "k>3", "score": 0.9, "index": 60}, {"type": "text", "coordinates": [472, 207, 541, 221], "content": "), using (3.2)", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [71, 221, 98, 235], "content": "with ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [98, 222, 127, 231], "content": "\\lambda=0", "score": 0.9, "index": 63}, {"type": "text", "coordinates": [128, 221, 200, 235], "content": ", tells us that ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [201, 222, 320, 233], "content": "2s+1=k,2r+1=m", "score": 0.47, "index": 65}, {"type": "text", "coordinates": [320, 221, 543, 235], "content": ". Now count the number of fixed-points of", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [71, 237, 79, 246], "content": "J", "score": 0.89, "index": 67}, {"type": "text", "coordinates": [80, 227, 159, 258], "content": " in both cases: ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [160, 234, 255, 250], "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [255, 227, 281, 258], "content": ", i.e. ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [281, 235, 372, 249], "content": "s-1=(k-1)/2", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [373, 227, 467, 258], "content": ", a contradiction.", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [93, 248, 542, 266], "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [70, 264, 326, 280], "content": "the number of weights appearing in the fusion ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [326, 264, 341, 277], "content": "\\Lambda_{\\star}", "score": 0.77, "index": 75}, {"type": "text", "coordinates": [342, 264, 358, 280], "content": "\u00d7 ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [358, 264, 373, 277], "content": "\\Lambda_{\\star}", "score": 0.73, "index": 77}, {"type": "text", "coordinates": [374, 264, 415, 280], "content": "(when ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [415, 264, 429, 277], "content": "\\Lambda_{\\star}", "score": 0.86, "index": 79}, {"type": "text", "coordinates": [430, 264, 541, 280], "content": "has second smallest", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [70, 278, 237, 294], "content": "q-dimension). For example, for ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [238, 280, 261, 293], "content": "A_{1,k}", "score": 0.84, "index": 82}, {"type": "text", "coordinates": [261, 278, 267, 294], "content": " ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [267, 278, 302, 292], "content": "\\left(k>1\\right)", "score": 0.43, "index": 84}, {"type": "text", "coordinates": [302, 278, 309, 294], "content": ", ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [309, 278, 332, 293], "content": "C_{r,k}", "score": 0.69, "index": 86}, {"type": "text", "coordinates": [333, 278, 339, 294], "content": " (", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [340, 279, 370, 291], "content": "k>1", "score": 0.82, "index": 88}, {"type": "text", "coordinates": [370, 278, 433, 294], "content": ", except for ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [433, 280, 513, 293], "content": "C_{2,2},C_{2,3},C_{3,2})", "score": 0.92, "index": 90}, {"type": "text", "coordinates": [513, 278, 542, 294], "content": ", and", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [71, 295, 93, 307], "content": "E_{7,k}", "score": 0.86, "index": 92}, {"type": "text", "coordinates": [94, 294, 101, 308], "content": " (", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [101, 294, 133, 306], "content": "k>4)", "score": 0.51, "index": 94}, {"type": "text", "coordinates": [133, 294, 541, 308], "content": "), we learned in \u00a73 that this number is 2, 3, 4 respectively, so none of these can", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [70, 308, 148, 322], "content": "be isomorphic.", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [95, 323, 541, 337], "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [70, 338, 254, 353], "content": "smallest q-dimension (respectively ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [254, 342, 260, 348], "content": "r", "score": 0.87, "index": 99}, {"type": "text", "coordinates": [260, 338, 286, 353], "content": " and ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [286, 337, 313, 349], "content": "r-1", "score": 0.86, "index": 101}, {"type": "text", "coordinates": [314, 338, 335, 353], "content": " for ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [335, 338, 357, 351], "content": "B_{r,2}", "score": 0.91, "index": 103}, {"type": "text", "coordinates": [357, 338, 383, 353], "content": " and ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [384, 339, 407, 351], "content": "D_{r,2}", "score": 0.92, "index": 105}, {"type": "text", "coordinates": [407, 338, 469, 353], "content": ", except for ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [470, 339, 492, 351], "content": "D_{4,2}", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [493, 338, 502, 353], "content": ").", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [93, 352, 308, 367], "content": "For the exceptional algebras, comparing ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [309, 352, 342, 366], "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "score": 0.92, "index": 110}, {"type": "text", "coordinates": [342, 352, 541, 367], "content": " and the number of highest-weights is", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [70, 365, 210, 383], "content": "effective. Recall that both ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [210, 368, 237, 380], "content": "\|\|P_{+}\|\|", "score": 0.94, "index": 113}, {"type": "text", "coordinates": [237, 365, 263, 383], "content": "and ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [263, 368, 296, 380], "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [296, 365, 541, 383], "content": " for a fixed algebra monotonically increase with", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [71, 383, 78, 392], "content": "k", "score": 0.86, "index": 117}, {"type": "text", "coordinates": [78, 381, 169, 396], "content": " to (respectively) ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [170, 386, 182, 392], "content": "\\infty", "score": 0.86, "index": 119}, {"type": "text", "coordinates": [182, 381, 326, 396], "content": "and the Weyl dimension of ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [327, 383, 341, 393], "content": "\\Lambda_{\\star}", "score": 0.91, "index": 121}, {"type": "text", "coordinates": [341, 381, 486, 396], "content": ", which is 7, 26, and 248 for ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [487, 383, 540, 394], "content": "G_{2},F_{4},E_{8}", "score": 0.93, "index": 123}, {"type": "text", "coordinates": [70, 396, 163, 411], "content": "respectively. For ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [163, 397, 186, 410], "content": "E_{8,k}", "score": 0.92, "index": 125}, {"type": "text", "coordinates": [186, 396, 193, 411], "content": ", ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [194, 397, 226, 409], "content": "\\mathcal{D}(\\Lambda_{1})", "score": 0.93, "index": 127}, {"type": "text", "coordinates": [226, 396, 302, 411], "content": " exceeds 7 for ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [303, 397, 333, 408], "content": "k\\geq5", "score": 0.91, "index": 129}, {"type": "text", "coordinates": [333, 396, 442, 411], "content": ", and exceeds 26 for ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [442, 397, 479, 408], "content": "k\\geq11", "score": 0.88, "index": 131}, {"type": "text", "coordinates": [479, 396, 518, 411], "content": ", while ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [518, 397, 539, 409], "content": "F_{4,k}", "score": 0.92, "index": 133}, {"type": "text", "coordinates": [70, 409, 144, 425], "content": "exceeds 7 for ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [144, 412, 175, 422], "content": "k\\geq4", "score": 0.92, "index": 135}, {"type": "text", "coordinates": [175, 409, 366, 425], "content": ". The number of highest-weights of ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [367, 411, 421, 424], "content": "E_{8,4},E_{8,10}", "score": 0.92, "index": 137}, {"type": "text", "coordinates": [421, 409, 452, 425], "content": ", and ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [452, 412, 473, 424], "content": "F_{4,3}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [473, 409, 541, 425], "content": " are 10, 135,", "score": 1.0, "index": 140}, {"type": "text", "coordinates": [70, 424, 415, 438], "content": "and 9, so only a small number of possibilities need be considered.", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [270, 457, 342, 469], "content": "References", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [79, 481, 539, 495], "content": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. 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Geom. 3 (1994), 347\u2013374.", "score": 1.0, "index": 158}]	[]	[{"type": "inline", "coordinates": [106, 75, 120, 87], "content": "[x]", "caption": ""}, {"type": "inline", "coordinates": [381, 78, 388, 84], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [357, 90, 398, 100], "content": "\\ell\\,\\equiv\\,\\pm1", "caption": ""}, {"type": "inline", "coordinates": [437, 90, 450, 99], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [213, 104, 246, 115], "content": "k\\ \\leq\\ 2", "caption": ""}, {"type": "inline", "coordinates": [350, 104, 462, 116], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "caption": ""}, {"type": "inline", "coordinates": [178, 118, 216, 128], "content": "\\ell\\equiv\\pm1", "caption": ""}, {"type": "inline", "coordinates": [252, 119, 266, 127], "content": "2\\kappa", "caption": ""}, {"type": "inline", "coordinates": [347, 118, 357, 127], "content": "\\kappa^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [470, 118, 507, 128], "content": "\\ell\\equiv\\pm1", "caption": ""}, {"type": "inline", "coordinates": [71, 132, 88, 142], "content": "2\\kappa^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [244, 132, 277, 142], "content": "\\kappa=\\kappa^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [302, 133, 339, 142], "content": "X=B", "caption": ""}, {"type": "inline", "coordinates": [357, 133, 367, 142], "content": "D", "caption": ""}, {"type": "inline", "coordinates": [394, 133, 423, 142], "content": "k>2", "caption": ""}, {"type": "inline", "coordinates": [444, 133, 480, 142], "content": "X=C", "caption": ""}, {"type": "inline", "coordinates": [529, 133, 536, 142], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [107, 147, 210, 160], "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "caption": ""}, {"type": "inline", "coordinates": [403, 148, 521, 158], "content": "r+k+1=s+m+1", "caption": ""}, {"type": "inline", "coordinates": [270, 161, 345, 177], "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "caption": ""}, {"type": "inline", "coordinates": [485, 178, 542, 191], "content": "\\mathcal{R}(B_{r,k})\\cong", "caption": ""}, {"type": "inline", "coordinates": [71, 193, 117, 205], "content": "\\mathcal{R}(B_{s,m})", "caption": ""}, {"type": "inline", "coordinates": [144, 194, 212, 205], "content": "B_{r,k}\\ne B_{s,m}", "caption": ""}, {"type": "inline", "coordinates": [271, 193, 320, 205], "content": "k,m\\,>\\,2", "caption": ""}, {"type": "inline", "coordinates": [444, 194, 536, 204], "content": "2r+k\\,=\\,2s\\,+\\,m", "caption": ""}, {"type": "inline", "coordinates": [198, 207, 231, 220], "content": "\\mathcal{D}(\\Lambda_{1})", "caption": ""}, {"type": "inline", "coordinates": [443, 208, 472, 217], "content": "k>3", "caption": ""}, {"type": "inline", "coordinates": [98, 222, 127, 231], "content": "\\lambda=0", "caption": 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The absolute value of each of these is quickly seen to be greater than 1 unless $\\ell\\,\\equiv\\,\\pm1$ (mod $2\\kappa$ ), except for the orthogonal algebras when $k\\ \\leq\\ 2$ . An isomorphism $\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})$ would require then that whenever $\\ell\\equiv\\pm1$ (mod $2\\kappa$ ) is coprime to $\\kappa^{\\prime}$ , it must also satisfy $\\ell\\equiv\\pm1$ (mod $2\\kappa^{\\prime}$ ), and conversely. This forces $\\kappa=\\kappa^{\\prime}$ , for $X=B$ or $D$ and $k>2$ , or $X=C$ and any $k$ . ", "page_idx": 21}, {"type": "text", "text": "If $\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})$ , then that Galois argument implies $r+k+1=s+m+1$ , so compare numbers of highest-weights: $\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)$ . ", "page_idx": 21}, {"type": "text", "text": "A similar argument works for the orthogonal algebras. For instance suppose $\\mathcal{R}(B_{r,k})\\cong$ $\\mathcal{R}(B_{s,m})$ but $B_{r,k}\\ne B_{s,m}$ , and that $k,m\\,>\\,2$ . Then Galois implies $2r+k\\,=\\,2s\\,+\\,m$ . Comparing the value of $\\mathcal{D}(\\Lambda_{1})$ (the second smallest q-dimension when $k>3$ ), using (3.2) with $\\lambda=0$ , tells us that $2s+1=k,2r+1=m$ . Now count the number of fixed-points of $J$ in both cases: $\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}$ , i.e. $s-1=(k-1)/2$ , a contradiction. ", "page_idx": 21}, {"type": "text", "text": "For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $\\Lambda_{\\star}$ \u00d7 $\\Lambda_{\\star}$ (when $\\Lambda_{\\star}$ has second smallest q-dimension). For example, for $A_{1,k}$ $\\left(k>1\\right)$ , $C_{r,k}$ ( $k>1$ , except for $C_{2,2},C_{2,3},C_{3,2})$ , and $E_{7,k}$ ( $k>4)$ ), we learned in \u00a73 that this number is 2, 3, 4 respectively, so none of these can be isomorphic. ", "page_idx": 21}, {"type": "text", "text": "For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $r$ and $r-1$ for $B_{r,2}$ and $D_{r,2}$ , except for $D_{4,2}$ ). ", "page_idx": 21}, {"type": "text", "text": "For the exceptional algebras, comparing $\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$ and the number of highest-weights is effective. Recall that both $\|\|P_{+}\|\|$ and $\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$ for a fixed algebra monotonically increase with $k$ to (respectively) $\\infty$ and the Weyl dimension of $\\Lambda_{\\star}$ , which is 7, 26, and 248 for $G_{2},F_{4},E_{8}$ respectively. For $E_{8,k}$ , $\\mathcal{D}(\\Lambda_{1})$ exceeds 7 for $k\\geq5$ , and exceeds 26 for $k\\geq11$ , while $F_{4,k}$ exceeds 7 for $k\\geq4$ . The number of highest-weights of $E_{8,4},E_{8,10}$ , and $F_{4,3}$ are 10, 135, and 9, so only a small number of possibilities need be considered. ", "page_idx": 21}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 21}, {"type": "text", "text": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the classification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998), \n163\u2013166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99\u2013136. \n2. D. Bernard, String characters from Kac\u2013Moody automorphisms, Nucl. Phys. B288 (1987), 628\u2013648. \n3. J. B\u00a8ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999. \n4. A. Coste and T. 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The absolute value of each", "type": "text"}], "index": 0}, {"bbox": [70, 88, 541, 104], "spans": [{"bbox": [70, 88, 356, 104], "score": 1.0, "content": "of these is quickly seen to be greater than 1 unless ", "type": "text"}, {"bbox": [357, 90, 398, 100], "score": 0.91, "content": "\\ell\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [399, 88, 437, 104], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [437, 90, 450, 99], "score": 0.72, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [451, 88, 541, 104], "score": 1.0, "content": "), except for the", "type": "text"}], "index": 1}, {"bbox": [71, 102, 541, 118], "spans": [{"bbox": [71, 102, 212, 118], "score": 1.0, "content": "orthogonal algebras when ", "type": "text"}, {"bbox": [213, 104, 246, 115], "score": 0.92, "content": "k\\ \\leq\\ 2", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [246, 102, 349, 118], "score": 1.0, "content": ". An isomorphism ", "type": "text"}, {"bbox": [350, 104, 462, 116], "score": 0.94, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "type": "inline_equation", "height": 12, "width": 112}, {"bbox": [462, 102, 541, 118], "score": 1.0, "content": " would require", "type": "text"}], "index": 2}, {"bbox": [71, 117, 540, 131], "spans": [{"bbox": [71, 117, 177, 131], "score": 1.0, "content": "then that whenever ", "type": "text"}, {"bbox": [178, 118, 216, 128], "score": 0.9, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [216, 117, 252, 131], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [252, 119, 266, 127], "score": 0.66, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [266, 117, 347, 131], "score": 1.0, "content": ") is coprime to ", "type": "text"}, {"bbox": [347, 118, 357, 127], "score": 0.89, "content": "\\kappa^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 117, 469, 131], "score": 1.0, "content": ", it must also satisfy ", "type": "text"}, {"bbox": [470, 118, 507, 128], "score": 0.91, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [508, 117, 540, 131], "score": 1.0, "content": " (mod", "type": "text"}], "index": 3}, {"bbox": [71, 131, 540, 145], "spans": [{"bbox": [71, 132, 88, 142], "score": 0.77, "content": "2\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [88, 131, 243, 145], "score": 1.0, "content": "), and conversely. This forces ", "type": "text"}, {"bbox": [244, 132, 277, 142], "score": 0.91, "content": "\\kappa=\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [277, 131, 302, 145], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [302, 133, 339, 142], "score": 0.92, "content": "X=B", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [339, 131, 356, 145], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [357, 133, 367, 142], "score": 0.91, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 131, 393, 145], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 133, 423, 142], "score": 0.91, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [423, 131, 443, 145], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [444, 133, 480, 142], "score": 0.92, "content": "X=C", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [480, 131, 529, 145], "score": 1.0, "content": " and any ", "type": "text"}, {"bbox": [529, 133, 536, 142], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [537, 131, 540, 145], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [70, 144, 541, 175], "lines": [{"bbox": [94, 146, 542, 163], "spans": [{"bbox": [94, 146, 107, 163], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 147, 210, 160], "score": 0.93, "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [210, 146, 403, 163], "score": 1.0, "content": ", then that Galois argument implies ", "type": "text"}, {"bbox": [403, 148, 521, 158], "score": 0.92, "content": "r+k+1=s+m+1", "type": "inline_equation", "height": 10, "width": 118}, {"bbox": [522, 146, 542, 163], "score": 1.0, "content": ", so", "type": "text"}], "index": 5}, {"bbox": [70, 160, 349, 179], "spans": [{"bbox": [70, 160, 269, 179], "score": 1.0, "content": "compare numbers of highest-weights: ", "type": "text"}, {"bbox": [270, 161, 345, 177], "score": 0.96, "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [345, 160, 349, 179], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [70, 176, 541, 248], "lines": [{"bbox": [94, 176, 542, 193], "spans": [{"bbox": [94, 176, 485, 193], "score": 1.0, "content": "A similar argument works for the orthogonal algebras. For instance suppose ", "type": "text"}, {"bbox": [485, 178, 542, 191], "score": 0.93, "content": "\\mathcal{R}(B_{r,k})\\cong", "type": "inline_equation", "height": 13, "width": 57}], "index": 7}, {"bbox": [71, 191, 541, 207], "spans": [{"bbox": [71, 193, 117, 205], "score": 0.93, "content": "\\mathcal{R}(B_{s,m})", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [117, 191, 144, 207], "score": 1.0, "content": " but ", "type": "text"}, {"bbox": [144, 194, 212, 205], "score": 0.93, "content": "B_{r,k}\\ne B_{s,m}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [212, 191, 271, 207], "score": 1.0, "content": ", and that ", "type": "text"}, {"bbox": [271, 193, 320, 205], "score": 0.89, "content": "k,m\\,>\\,2", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [321, 191, 443, 207], "score": 1.0, "content": ". Then Galois implies ", "type": "text"}, {"bbox": [444, 194, 536, 204], "score": 0.91, "content": "2r+k\\,=\\,2s\\,+\\,m", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [536, 191, 541, 207], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [72, 207, 541, 221], "spans": [{"bbox": [72, 207, 198, 221], "score": 1.0, "content": "Comparing the value of ", "type": "text"}, {"bbox": [198, 207, 231, 220], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [231, 207, 442, 221], "score": 1.0, "content": " (the second smallest q-dimension when ", "type": "text"}, {"bbox": [443, 208, 472, 217], "score": 0.9, "content": "k>3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [472, 207, 541, 221], "score": 1.0, "content": "), using (3.2)", "type": "text"}], "index": 9}, {"bbox": [71, 221, 543, 235], "spans": [{"bbox": [71, 221, 98, 235], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [98, 222, 127, 231], "score": 0.9, "content": "\\lambda=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [128, 221, 200, 235], "score": 1.0, "content": ", tells us that ", "type": "text"}, {"bbox": [201, 222, 320, 233], "score": 0.47, "content": "2s+1=k,2r+1=m", "type": "inline_equation", "height": 11, "width": 119}, {"bbox": [320, 221, 543, 235], "score": 1.0, "content": ". Now count the number of fixed-points of", "type": "text"}], "index": 10}, {"bbox": [71, 227, 467, 258], "spans": [{"bbox": [71, 237, 79, 246], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [80, 227, 159, 258], "score": 1.0, "content": " in both cases: ", "type": "text"}, {"bbox": [160, 234, 255, 250], "score": 0.93, "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "type": "inline_equation", "height": 16, "width": 95}, {"bbox": [255, 227, 281, 258], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [281, 235, 372, 249], "score": 0.92, "content": "s-1=(k-1)/2", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [373, 227, 467, 258], "score": 1.0, "content": ", a contradiction.", "type": "text"}], "index": 11}], "index": 9}, {"type": "text", "bbox": [71, 249, 541, 319], "lines": [{"bbox": [93, 248, 542, 266], "spans": [{"bbox": [93, 248, 542, 266], "score": 1.0, "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "type": "text"}], "index": 12}, {"bbox": [70, 264, 541, 280], "spans": [{"bbox": [70, 264, 326, 280], "score": 1.0, "content": "the number of weights appearing in the fusion ", "type": "text"}, {"bbox": [326, 264, 341, 277], "score": 0.77, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [342, 264, 358, 280], "score": 1.0, "content": "\u00d7 ", "type": "text"}, {"bbox": [358, 264, 373, 277], "score": 0.73, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [374, 264, 415, 280], "score": 1.0, "content": "(when ", "type": "text"}, {"bbox": [415, 264, 429, 277], "score": 0.86, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [430, 264, 541, 280], "score": 1.0, "content": "has second smallest", "type": "text"}], "index": 13}, {"bbox": [70, 278, 542, 294], "spans": [{"bbox": [70, 278, 237, 294], "score": 1.0, "content": "q-dimension). For example, for ", "type": "text"}, {"bbox": [238, 280, 261, 293], "score": 0.84, "content": "A_{1,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [261, 278, 267, 294], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [267, 278, 302, 292], "score": 0.43, "content": "\\left(k>1\\right)", "type": "inline_equation", "height": 14, "width": 35}, {"bbox": [302, 278, 309, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [309, 278, 332, 293], "score": 0.69, "content": "C_{r,k}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [333, 278, 339, 294], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [340, 279, 370, 291], "score": 0.82, "content": "k>1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [370, 278, 433, 294], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [433, 280, 513, 293], "score": 0.92, "content": "C_{2,2},C_{2,3},C_{3,2})", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [513, 278, 542, 294], "score": 1.0, "content": ", and", "type": "text"}], "index": 14}, {"bbox": [71, 294, 541, 308], "spans": [{"bbox": [71, 295, 93, 307], "score": 0.86, "content": "E_{7,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [94, 294, 101, 308], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [101, 294, 133, 306], "score": 0.51, "content": "k>4)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [133, 294, 541, 308], "score": 1.0, "content": "), we learned in \u00a73 that this number is 2, 3, 4 respectively, so none of these can", "type": "text"}], "index": 15}, {"bbox": [70, 308, 148, 322], "spans": [{"bbox": [70, 308, 148, 322], "score": 1.0, "content": "be isomorphic.", "type": "text"}], "index": 16}], "index": 14}, {"type": "text", "bbox": [70, 321, 541, 349], "lines": [{"bbox": [95, 323, 541, 337], "spans": [{"bbox": [95, 323, 541, 337], "score": 1.0, "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "type": "text"}], "index": 17}, {"bbox": [70, 337, 502, 353], "spans": [{"bbox": [70, 338, 254, 353], "score": 1.0, "content": "smallest q-dimension (respectively ", "type": "text"}, {"bbox": [254, 342, 260, 348], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [260, 338, 286, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 337, 313, 349], "score": 0.86, "content": "r-1", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [314, 338, 335, 353], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [335, 338, 357, 351], "score": 0.91, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [357, 338, 383, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [384, 339, 407, 351], "score": 0.92, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [407, 338, 469, 353], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [470, 339, 492, 351], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [493, 338, 502, 353], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [70, 351, 541, 436], "lines": [{"bbox": [93, 352, 541, 367], "spans": [{"bbox": [93, 352, 308, 367], "score": 1.0, "content": "For the exceptional algebras, comparing ", "type": "text"}, {"bbox": [309, 352, 342, 366], "score": 0.92, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [342, 352, 541, 367], "score": 1.0, "content": " and the number of highest-weights is", "type": "text"}], "index": 19}, {"bbox": [70, 365, 541, 383], "spans": [{"bbox": [70, 365, 210, 383], "score": 1.0, "content": "effective. Recall that both ", "type": "text"}, {"bbox": [210, 368, 237, 380], "score": 0.94, "content": "\|\|P_{+}\|\|", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [237, 365, 263, 383], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [263, 368, 296, 380], "score": 0.94, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [296, 365, 541, 383], "score": 1.0, "content": " for a fixed algebra monotonically increase with", "type": "text"}], "index": 20}, {"bbox": [71, 381, 540, 396], "spans": [{"bbox": [71, 383, 78, 392], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 381, 169, 396], "score": 1.0, "content": " to (respectively) ", "type": "text"}, {"bbox": [170, 386, 182, 392], "score": 0.86, "content": "\\infty", "type": "inline_equation", "height": 6, "width": 12}, {"bbox": [182, 381, 326, 396], "score": 1.0, "content": "and the Weyl dimension of ", "type": "text"}, {"bbox": [327, 383, 341, 393], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [341, 381, 486, 396], "score": 1.0, "content": ", which is 7, 26, and 248 for ", "type": "text"}, {"bbox": [487, 383, 540, 394], "score": 0.93, "content": "G_{2},F_{4},E_{8}", "type": "inline_equation", "height": 11, "width": 53}], "index": 21}, {"bbox": [70, 396, 539, 411], "spans": [{"bbox": [70, 396, 163, 411], "score": 1.0, "content": "respectively. For ", "type": "text"}, {"bbox": [163, 397, 186, 410], "score": 0.92, "content": "E_{8,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [186, 396, 193, 411], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [194, 397, 226, 409], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [226, 396, 302, 411], "score": 1.0, "content": " exceeds 7 for ", "type": "text"}, {"bbox": [303, 397, 333, 408], "score": 0.91, "content": "k\\geq5", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [333, 396, 442, 411], "score": 1.0, "content": ", and exceeds 26 for ", "type": "text"}, {"bbox": [442, 397, 479, 408], "score": 0.88, "content": "k\\geq11", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [479, 396, 518, 411], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [518, 397, 539, 409], "score": 0.92, "content": "F_{4,k}", "type": "inline_equation", "height": 12, "width": 21}], "index": 22}, {"bbox": [70, 409, 541, 425], "spans": [{"bbox": [70, 409, 144, 425], "score": 1.0, "content": "exceeds 7 for ", "type": "text"}, {"bbox": [144, 412, 175, 422], "score": 0.92, "content": "k\\geq4", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [175, 409, 366, 425], "score": 1.0, "content": ". The number of highest-weights of ", "type": "text"}, {"bbox": [367, 411, 421, 424], "score": 0.92, "content": "E_{8,4},E_{8,10}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [421, 409, 452, 425], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [452, 412, 473, 424], "score": 0.92, "content": "F_{4,3}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [473, 409, 541, 425], "score": 1.0, "content": " are 10, 135,", "type": "text"}], "index": 23}, {"bbox": [70, 424, 415, 438], "spans": [{"bbox": [70, 424, 415, 438], "score": 1.0, "content": "and 9, so only a small number of possibilities need be considered.", "type": "text"}], "index": 24}], "index": 21.5}, {"type": "title", "bbox": [270, 455, 342, 468], "lines": [{"bbox": [270, 457, 342, 469], "spans": [{"bbox": [270, 457, 342, 469], "score": 1.0, "content": "References", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [78, 478, 541, 716], "lines": [{"bbox": [79, 481, 539, 495], "spans": [{"bbox": [79, 481, 539, 495], "score": 1.0, "content": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas-", "type": "text"}], "index": 26}, {"bbox": [94, 495, 540, 510], "spans": [{"bbox": [94, 495, 540, 510], "score": 1.0, "content": "sification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998),", "type": "text"}], "index": 27}, {"bbox": [95, 511, 142, 523], "spans": [{"bbox": [95, 511, 142, 523], "score": 1.0, "content": "163\u2013166;", "type": "text"}], "index": 28}, {"bbox": [95, 525, 539, 539], "spans": [{"bbox": [95, 525, 539, 539], "score": 1.0, "content": "J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds,", "type": "text"}], "index": 29}, {"bbox": [95, 539, 276, 554], "spans": [{"bbox": [95, 539, 276, 554], "score": 1.0, "content": "Nucl. Phys. B530 (1998), 99\u2013136.", "type": "text"}], "index": 30}, {"bbox": [78, 554, 541, 570], "spans": [{"bbox": [78, 554, 541, 570], "score": 1.0, "content": "2. D. Bernard, String characters from Kac\u2013Moody automorphisms, Nucl. Phys. B288", "type": "text"}], "index": 31}, {"bbox": [95, 570, 182, 583], "spans": [{"bbox": [95, 570, 182, 583], "score": 1.0, "content": "(1987), 628\u2013648.", "type": "text"}], "index": 32}, {"bbox": [78, 583, 542, 600], "spans": [{"bbox": [78, 583, 542, 600], "score": 1.0, "content": "3. J. B\u00a8ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I", "type": "text"}], "index": 33}, {"bbox": [94, 598, 523, 615], "spans": [{"bbox": [94, 598, 523, 615], "score": 1.0, "content": "coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999.", "type": "text"}], "index": 34}, {"bbox": [78, 612, 541, 630], "spans": [{"bbox": [78, 612, 541, 630], "score": 1.0, "content": "4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories,", "type": "text"}], "index": 35}, {"bbox": [95, 627, 279, 644], "spans": [{"bbox": [95, 627, 279, 644], "score": 1.0, "content": "Phys. Lett. B323 (1994), 316\u2013321.", "type": "text"}], "index": 36}, {"bbox": [78, 642, 540, 659], "spans": [{"bbox": [78, 642, 540, 659], "score": 1.0, "content": "5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hep-", "type": "text"}], "index": 37}, {"bbox": [95, 658, 190, 672], "spans": [{"bbox": [95, 658, 190, 672], "score": 1.0, "content": "th/0001158, 2000.", "type": "text"}], "index": 38}, {"bbox": [78, 672, 541, 689], "spans": [{"bbox": [78, 672, 541, 689], "score": 1.0, "content": "6. Ph. Di Francesco, P. Mathieu and D. S\u00b4en\u00b4echal, \u201cConformal Field Theory\u201d,", "type": "text"}], "index": 39}, {"bbox": [95, 689, 271, 702], "spans": [{"bbox": [95, 689, 271, 702], "score": 1.0, "content": "Springer-Verlag, New York, 1997.", "type": "text"}], "index": 40}, {"bbox": [79, 702, 525, 718], "spans": [{"bbox": [79, 702, 525, 718], "score": 1.0, "content": "7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347\u2013374.", "type": "text"}], "index": 41}], "index": 33.5}], "layout_bboxes": [], "page_idx": 21, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 314, 744], "spans": [{"bbox": [298, 731, 314, 744], "score": 1.0, "content": "22", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 541, 143], "lines": [{"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 106, 89], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 75, 120, 87], "score": 0.9, "content": "[x]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [120, 73, 381, 89], "score": 1.0, "content": " here denotes the greatest integer not more than ", "type": "text"}, {"bbox": [381, 78, 388, 84], "score": 0.89, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [389, 73, 541, 89], "score": 1.0, "content": ". The absolute value of each", "type": "text"}], "index": 0}, {"bbox": [70, 88, 541, 104], "spans": [{"bbox": [70, 88, 356, 104], "score": 1.0, "content": "of these is quickly seen to be greater than 1 unless ", "type": "text"}, {"bbox": [357, 90, 398, 100], "score": 0.91, "content": "\\ell\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [399, 88, 437, 104], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [437, 90, 450, 99], "score": 0.72, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [451, 88, 541, 104], "score": 1.0, "content": "), except for the", "type": "text"}], "index": 1}, {"bbox": [71, 102, 541, 118], "spans": [{"bbox": [71, 102, 212, 118], "score": 1.0, "content": "orthogonal algebras when ", "type": "text"}, {"bbox": [213, 104, 246, 115], "score": 0.92, "content": "k\\ \\leq\\ 2", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [246, 102, 349, 118], "score": 1.0, "content": ". An isomorphism ", "type": "text"}, {"bbox": [350, 104, 462, 116], "score": 0.94, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "type": "inline_equation", "height": 12, "width": 112}, {"bbox": [462, 102, 541, 118], "score": 1.0, "content": " would require", "type": "text"}], "index": 2}, {"bbox": [71, 117, 540, 131], "spans": [{"bbox": [71, 117, 177, 131], "score": 1.0, "content": "then that whenever ", "type": "text"}, {"bbox": [178, 118, 216, 128], "score": 0.9, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [216, 117, 252, 131], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [252, 119, 266, 127], "score": 0.66, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [266, 117, 347, 131], "score": 1.0, "content": ") is coprime to ", "type": "text"}, {"bbox": [347, 118, 357, 127], "score": 0.89, "content": "\\kappa^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 117, 469, 131], "score": 1.0, "content": ", it must also satisfy ", "type": "text"}, {"bbox": [470, 118, 507, 128], "score": 0.91, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [508, 117, 540, 131], "score": 1.0, "content": " (mod", "type": "text"}], "index": 3}, {"bbox": [71, 131, 540, 145], "spans": [{"bbox": [71, 132, 88, 142], "score": 0.77, "content": "2\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [88, 131, 243, 145], "score": 1.0, "content": "), and conversely. This forces ", "type": "text"}, {"bbox": [244, 132, 277, 142], "score": 0.91, "content": "\\kappa=\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [277, 131, 302, 145], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [302, 133, 339, 142], "score": 0.92, "content": "X=B", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [339, 131, 356, 145], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [357, 133, 367, 142], "score": 0.91, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 131, 393, 145], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 133, 423, 142], "score": 0.91, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [423, 131, 443, 145], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [444, 133, 480, 142], "score": 0.92, "content": "X=C", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [480, 131, 529, 145], "score": 1.0, "content": " and any ", "type": "text"}, {"bbox": [529, 133, 536, 142], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [537, 131, 540, 145], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [70, 73, 541, 145]}, {"type": "text", "bbox": [70, 144, 541, 175], "lines": [{"bbox": [94, 146, 542, 163], "spans": [{"bbox": [94, 146, 107, 163], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 147, 210, 160], "score": 0.93, "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [210, 146, 403, 163], "score": 1.0, "content": ", then that Galois argument implies ", "type": "text"}, {"bbox": [403, 148, 521, 158], "score": 0.92, "content": "r+k+1=s+m+1", "type": "inline_equation", "height": 10, "width": 118}, {"bbox": [522, 146, 542, 163], "score": 1.0, "content": ", so", "type": "text"}], "index": 5}, {"bbox": [70, 160, 349, 179], "spans": [{"bbox": [70, 160, 269, 179], "score": 1.0, "content": "compare numbers of highest-weights: ", "type": "text"}, {"bbox": [270, 161, 345, 177], "score": 0.96, "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [345, 160, 349, 179], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [70, 146, 542, 179]}, {"type": "text", "bbox": [70, 176, 541, 248], "lines": [{"bbox": [94, 176, 542, 193], "spans": [{"bbox": [94, 176, 485, 193], "score": 1.0, "content": "A similar argument works for the orthogonal algebras. For instance suppose ", "type": "text"}, {"bbox": [485, 178, 542, 191], "score": 0.93, "content": "\\mathcal{R}(B_{r,k})\\cong", "type": "inline_equation", "height": 13, "width": 57}], "index": 7}, {"bbox": [71, 191, 541, 207], "spans": [{"bbox": [71, 193, 117, 205], "score": 0.93, "content": "\\mathcal{R}(B_{s,m})", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [117, 191, 144, 207], "score": 1.0, "content": " but ", "type": "text"}, {"bbox": [144, 194, 212, 205], "score": 0.93, "content": "B_{r,k}\\ne B_{s,m}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [212, 191, 271, 207], "score": 1.0, "content": ", and that ", "type": "text"}, {"bbox": [271, 193, 320, 205], "score": 0.89, "content": "k,m\\,>\\,2", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [321, 191, 443, 207], "score": 1.0, "content": ". Then Galois implies ", "type": "text"}, {"bbox": [444, 194, 536, 204], "score": 0.91, "content": "2r+k\\,=\\,2s\\,+\\,m", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [536, 191, 541, 207], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [72, 207, 541, 221], "spans": [{"bbox": [72, 207, 198, 221], "score": 1.0, "content": "Comparing the value of ", "type": "text"}, {"bbox": [198, 207, 231, 220], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [231, 207, 442, 221], "score": 1.0, "content": " (the second smallest q-dimension when ", "type": "text"}, {"bbox": [443, 208, 472, 217], "score": 0.9, "content": "k>3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [472, 207, 541, 221], "score": 1.0, "content": "), using (3.2)", "type": "text"}], "index": 9}, {"bbox": [71, 221, 543, 235], "spans": [{"bbox": [71, 221, 98, 235], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [98, 222, 127, 231], "score": 0.9, "content": "\\lambda=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [128, 221, 200, 235], "score": 1.0, "content": ", tells us that ", "type": "text"}, {"bbox": [201, 222, 320, 233], "score": 0.47, "content": "2s+1=k,2r+1=m", "type": "inline_equation", "height": 11, "width": 119}, {"bbox": [320, 221, 543, 235], "score": 1.0, "content": ". Now count the number of fixed-points of", "type": "text"}], "index": 10}, {"bbox": [71, 227, 467, 258], "spans": [{"bbox": [71, 237, 79, 246], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [80, 227, 159, 258], "score": 1.0, "content": " in both cases: ", "type": "text"}, {"bbox": [160, 234, 255, 250], "score": 0.93, "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "type": "inline_equation", "height": 16, "width": 95}, {"bbox": [255, 227, 281, 258], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [281, 235, 372, 249], "score": 0.92, "content": "s-1=(k-1)/2", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [373, 227, 467, 258], "score": 1.0, "content": ", a contradiction.", "type": "text"}], "index": 11}], "index": 9, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [71, 176, 543, 258]}, {"type": "text", "bbox": [71, 249, 541, 319], "lines": [{"bbox": [93, 248, 542, 266], "spans": [{"bbox": [93, 248, 542, 266], "score": 1.0, "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "type": "text"}], "index": 12}, {"bbox": [70, 264, 541, 280], "spans": [{"bbox": [70, 264, 326, 280], "score": 1.0, "content": "the number of weights appearing in the fusion ", "type": "text"}, {"bbox": [326, 264, 341, 277], "score": 0.77, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [342, 264, 358, 280], "score": 1.0, "content": "\u00d7 ", "type": "text"}, {"bbox": [358, 264, 373, 277], "score": 0.73, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [374, 264, 415, 280], "score": 1.0, "content": "(when ", "type": "text"}, {"bbox": [415, 264, 429, 277], "score": 0.86, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [430, 264, 541, 280], "score": 1.0, "content": "has second smallest", "type": "text"}], "index": 13}, {"bbox": [70, 278, 542, 294], "spans": [{"bbox": [70, 278, 237, 294], "score": 1.0, "content": "q-dimension). For example, for ", "type": "text"}, {"bbox": [238, 280, 261, 293], "score": 0.84, "content": "A_{1,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [261, 278, 267, 294], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [267, 278, 302, 292], "score": 0.43, "content": "\\left(k>1\\right)", "type": "inline_equation", "height": 14, "width": 35}, {"bbox": [302, 278, 309, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [309, 278, 332, 293], "score": 0.69, "content": "C_{r,k}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [333, 278, 339, 294], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [340, 279, 370, 291], "score": 0.82, "content": "k>1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [370, 278, 433, 294], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [433, 280, 513, 293], "score": 0.92, "content": "C_{2,2},C_{2,3},C_{3,2})", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [513, 278, 542, 294], "score": 1.0, "content": ", and", "type": "text"}], "index": 14}, {"bbox": [71, 294, 541, 308], "spans": [{"bbox": [71, 295, 93, 307], "score": 0.86, "content": "E_{7,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [94, 294, 101, 308], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [101, 294, 133, 306], "score": 0.51, "content": "k>4)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [133, 294, 541, 308], "score": 1.0, "content": "), we learned in \u00a73 that this number is 2, 3, 4 respectively, so none of these can", "type": "text"}], "index": 15}, {"bbox": [70, 308, 148, 322], "spans": [{"bbox": [70, 308, 148, 322], "score": 1.0, "content": "be isomorphic.", "type": "text"}], "index": 16}], "index": 14, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [70, 248, 542, 322]}, {"type": "text", "bbox": [70, 321, 541, 349], "lines": [{"bbox": [95, 323, 541, 337], "spans": [{"bbox": [95, 323, 541, 337], "score": 1.0, "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "type": "text"}], "index": 17}, {"bbox": [70, 337, 502, 353], "spans": [{"bbox": [70, 338, 254, 353], "score": 1.0, "content": "smallest q-dimension (respectively ", "type": "text"}, {"bbox": [254, 342, 260, 348], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [260, 338, 286, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 337, 313, 349], "score": 0.86, "content": "r-1", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [314, 338, 335, 353], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [335, 338, 357, 351], "score": 0.91, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [357, 338, 383, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [384, 339, 407, 351], "score": 0.92, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [407, 338, 469, 353], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [470, 339, 492, 351], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [493, 338, 502, 353], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 17.5, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [70, 323, 541, 353]}, {"type": "text", "bbox": [70, 351, 541, 436], "lines": [{"bbox": [93, 352, 541, 367], "spans": [{"bbox": [93, 352, 308, 367], "score": 1.0, "content": "For the exceptional algebras, comparing ", "type": "text"}, {"bbox": [309, 352, 342, 366], "score": 0.92, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [342, 352, 541, 367], "score": 1.0, "content": " and the number of highest-weights is", "type": "text"}], "index": 19}, {"bbox": [70, 365, 541, 383], "spans": [{"bbox": [70, 365, 210, 383], "score": 1.0, "content": "effective. Recall that both ", "type": "text"}, {"bbox": [210, 368, 237, 380], "score": 0.94, "content": "\|\|P_{+}\|\|", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [237, 365, 263, 383], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [263, 368, 296, 380], "score": 0.94, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [296, 365, 541, 383], "score": 1.0, "content": " for a fixed algebra monotonically increase with", "type": "text"}], "index": 20}, {"bbox": [71, 381, 540, 396], "spans": [{"bbox": [71, 383, 78, 392], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 381, 169, 396], "score": 1.0, "content": " to (respectively) ", "type": "text"}, {"bbox": [170, 386, 182, 392], "score": 0.86, "content": "\\infty", "type": "inline_equation", "height": 6, "width": 12}, {"bbox": [182, 381, 326, 396], "score": 1.0, "content": "and the Weyl dimension of ", "type": "text"}, {"bbox": [327, 383, 341, 393], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [341, 381, 486, 396], "score": 1.0, "content": ", which is 7, 26, and 248 for ", "type": "text"}, {"bbox": [487, 383, 540, 394], "score": 0.93, "content": "G_{2},F_{4},E_{8}", "type": "inline_equation", "height": 11, "width": 53}], "index": 21}, {"bbox": [70, 396, 539, 411], "spans": [{"bbox": [70, 396, 163, 411], "score": 1.0, "content": "respectively. For ", "type": "text"}, {"bbox": [163, 397, 186, 410], "score": 0.92, "content": "E_{8,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [186, 396, 193, 411], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [194, 397, 226, 409], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [226, 396, 302, 411], "score": 1.0, "content": " exceeds 7 for ", "type": "text"}, {"bbox": [303, 397, 333, 408], "score": 0.91, "content": "k\\geq5", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [333, 396, 442, 411], "score": 1.0, "content": ", and exceeds 26 for ", "type": "text"}, {"bbox": [442, 397, 479, 408], "score": 0.88, "content": "k\\geq11", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [479, 396, 518, 411], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [518, 397, 539, 409], "score": 0.92, "content": "F_{4,k}", "type": "inline_equation", "height": 12, "width": 21}], "index": 22}, {"bbox": [70, 409, 541, 425], "spans": [{"bbox": [70, 409, 144, 425], "score": 1.0, "content": "exceeds 7 for ", "type": "text"}, {"bbox": [144, 412, 175, 422], "score": 0.92, "content": "k\\geq4", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [175, 409, 366, 425], "score": 1.0, "content": ". The number of highest-weights of ", "type": "text"}, {"bbox": [367, 411, 421, 424], "score": 0.92, "content": "E_{8,4},E_{8,10}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [421, 409, 452, 425], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [452, 412, 473, 424], "score": 0.92, "content": "F_{4,3}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [473, 409, 541, 425], "score": 1.0, "content": " are 10, 135,", "type": "text"}], "index": 23}, {"bbox": [70, 424, 415, 438], "spans": [{"bbox": [70, 424, 415, 438], "score": 1.0, "content": "and 9, so only a small number of possibilities need be considered.", "type": "text"}], "index": 24}], "index": 21.5, "page_num": "page_21", "page_size": [612.0, 792.0], "bbox_fs": [70, 352, 541, 438]}, {"type": "title", "bbox": [270, 455, 342, 468], "lines": [{"bbox": [270, 457, 342, 469], "spans": [{"bbox": [270, 457, 342, 469], "score": 1.0, "content": "References", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_21", "page_size": [612.0, 792.0]}, {"type": "list", "bbox": [78, 478, 541, 716], "lines": [{"bbox": [79, 481, 539, 495], "spans": [{"bbox": [79, 481, 539, 495], "score": 1.0, "content": "1. 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0002044v1	17	$$k=1$$ is easy: $$P_{+}=\{0,J0,\Lambda_{r}\}$$ and $$\pi=i d$$ . is automatic. $$k=2$$ will be done later in this subsection. Assume now that $$k\geq3$$ . From Proposition 4.1(b) we can write $$\pi\Lambda_{1}\,=\,J^{a}\Lambda_{1}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,J^{a^{\prime}}\Lambda_{1}$$ . We know $$\pi J0\,=\,J0$$ , so (2.7b) says $$\pi$$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $$\chi_{\Lambda_{1}}[\psi]\,=\,(-1)^{a^{\prime}}\chi_{\Lambda_{1}}[\pi\psi]$$ for any spinor $$\psi$$ . Now if $$a^{\prime}=1$$ , then $$\pi$$ will take the spinors which maximize $$\chi_{\Lambda_{1}}$$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $$\pi(S\Lambda_{r})$$ equals $$k\Lambda_{r}$$ (when $$k$$ odd) or $$S((k-1)\Lambda_{r})$$ (when $$k$$ even). But the sets $$S\Lambda_{r}$$ and $$k\Lambda_{r}$$ have different cardi- nalities ( $$k\Lambda_{r}$$ is a $$J$$ -fixed-point), and so can’t get mapped to each other. Also, the fusions $$\Lambda_{1}$$ × $$\Lambda_{r}=\Lambda_{r}$$ + $$(\Lambda_{1}+\Lambda_{r})$$ and $$J^{a}\Lambda_{1}\boxtimes\left(J^{i}(k-1)\Lambda_{r}\right)=\left(J^{a+i}(k-1)\Lambda_{r}\right)$$ + $$(J^{a+i+1}(k-$$ $$1)\Lambda_{r})$$ + $$J^{a+i+1}(\Lambda_{r-1}+(k-3)\Lambda_{r})$$ have different numbers of weights on their right sides, so also $$\pi\Lambda_{r}\notin{\cal S}(k-1)\Lambda_{r}$$ . Thus $$a^{\prime}=0$$ and $$\pi\Lambda_{r}=J^{b}\Lambda_{r}$$ for some $$b$$ . Similarly, $$a=0$$ . Hitting $$\pi$$ with $$\pi[1]^{b}$$ , we may assume that $$\pi$$ fixes $$\Lambda_{r}$$ . Now assume $$\pi$$ fixes $$\Lambda_{\ell}$$ , for $$1\leq\ell<r-1$$ . Then the fusion $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ says that $$\pi\Lambda_{\ell+1}$$ equals $$\Lambda_{\ell+1}$$ or $$\Lambda_{1}+\Lambda_{\ell}$$ . But from (3.2) we find Hence $$\pi$$ will fix $$\Lambda_{\ell+1}$$ if it fixes $$\Lambda_{\ell}$$ , concluding the argument. Now consider the more interesting case: $$k=2$$ . Then $$\kappa=2r+1$$ ; recall the weights in $$P_{+}(B_{r,2})$$ are the simple-currents $$0,J0$$ , the $$J$$ -fixed-points $$\gamma^{1},\ldots,\gamma^{r}$$ (notation defined in §3.2), and the spinors $$\Lambda_{r},J\Lambda_{r}$$ . Because $$\pi(J0)\,=\,\pi^{\prime}(J0)\,=\,J0$$ , we know both $$\pi$$ and $$\pi^{\prime}$$ must take $$J$$ -fixed-points to $$J$$ -fixed-points, i.e. $$\pi\Lambda_{1}\,=\,\gamma^{m}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,\gamma^{m^{\prime}}$$ for some $$1\leq m,m^{\prime}\leq r$$ . It is easy to compute [25] From this we see $$m\,m^{\prime}\equiv\pm1$$ (mod $$\kappa$$ ), so $$^{\prime\prime}$$ is coprime to $$\kappa$$ . Hitting it with the Galois fusion-symmetry $$\pi\{m^{\prime}\}$$ , we see that we may assume $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ . Now use (4.2) to get $$\pi\gamma^{i}=\gamma^{i}$$ for all $$i$$ . Then $$\pi$$ equals the identity or $$\pi[1]$$ , depending on what $$\pi$$ does to $$\Lambda_{r}$$ . # 4.4. The $$C$$ -series argument By rank-level duality, we may take $$r\le k$$ . For now assume $$(r,k)\neq(2,3)$$ . Then we know $$\pi\Lambda_{1}=J^{a}\Lambda_{1}$$ and $$\pi\Lambda_{1}=J^{a^{\prime}}\Lambda_{1}$$ for some $$a,a^{\prime}$$ . Since $$\pi J0=\pi^{\prime}J0=J0$$ , (2.7b) says $$a=a^{\prime}=0$$ if $$k r$$ is odd. Since $$\chi_{\Lambda_{1}}[\Lambda_{1}]>0$$ (using (3.3)), $$S_{\Lambda_{1}\Lambda_{1}}=S_{J^{a}\Lambda_{1},J^{a^{\prime}}\Lambda_{1}}$$ implies that $$a=a^{\prime}$$ also holds when $$k r$$ is even, and hence we may assume (hitting with $$\pi[1]^{a}$$ ) that also	<p>$$k=1$$ is easy: $$P_{+}=\{0,J0,\Lambda_{r}\}$$ and $$\pi=i d$$ . is automatic. $$k=2$$ will be done later in this subsection. Assume now that $$k\geq3$$ .</p> <p>From Proposition 4.1(b) we can write $$\pi\Lambda_{1}\,=\,J^{a}\Lambda_{1}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,J^{a^{\prime}}\Lambda_{1}$$ . We know $$\pi J0\,=\,J0$$ , so (2.7b) says $$\pi$$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $$\chi_{\Lambda_{1}}[\psi]\,=\,(-1)^{a^{\prime}}\chi_{\Lambda_{1}}[\pi\psi]$$ for any spinor $$\psi$$ . Now if $$a^{\prime}=1$$ , then $$\pi$$ will take the spinors which maximize $$\chi_{\Lambda_{1}}$$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $$\pi(S\Lambda_{r})$$ equals $$k\Lambda_{r}$$ (when $$k$$ odd) or $$S((k-1)\Lambda_{r})$$ (when $$k$$ even). But the sets $$S\Lambda_{r}$$ and $$k\Lambda_{r}$$ have different cardi- nalities ( $$k\Lambda_{r}$$ is a $$J$$ -fixed-point), and so can’t get mapped to each other. Also, the fusions $$\Lambda_{1}$$ × $$\Lambda_{r}=\Lambda_{r}$$ + $$(\Lambda_{1}+\Lambda_{r})$$ and $$J^{a}\Lambda_{1}\boxtimes\left(J^{i}(k-1)\Lambda_{r}\right)=\left(J^{a+i}(k-1)\Lambda_{r}\right)$$ + $$(J^{a+i+1}(k-$$ $$1)\Lambda_{r})$$ + $$J^{a+i+1}(\Lambda_{r-1}+(k-3)\Lambda_{r})$$ have different numbers of weights on their right sides, so also $$\pi\Lambda_{r}\notin{\cal S}(k-1)\Lambda_{r}$$ .</p> <p>Thus $$a^{\prime}=0$$ and $$\pi\Lambda_{r}=J^{b}\Lambda_{r}$$ for some $$b$$ . Similarly, $$a=0$$ . Hitting $$\pi$$ with $$\pi[1]^{b}$$ , we may assume that $$\pi$$ fixes $$\Lambda_{r}$$ .</p> <p>Now assume $$\pi$$ fixes $$\Lambda_{\ell}$$ , for $$1\leq\ell<r-1$$ . Then the fusion $$\Lambda_{1}$$ × $$\Lambda_{\ell}$$ says that $$\pi\Lambda_{\ell+1}$$ equals $$\Lambda_{\ell+1}$$ or $$\Lambda_{1}+\Lambda_{\ell}$$ . But from (3.2) we find</p> <p>Hence $$\pi$$ will fix $$\Lambda_{\ell+1}$$ if it fixes $$\Lambda_{\ell}$$ , concluding the argument.</p> <p>Now consider the more interesting case: $$k=2$$ . Then $$\kappa=2r+1$$ ; recall the weights in $$P_{+}(B_{r,2})$$ are the simple-currents $$0,J0$$ , the $$J$$ -fixed-points $$\gamma^{1},\ldots,\gamma^{r}$$ (notation defined in §3.2), and the spinors $$\Lambda_{r},J\Lambda_{r}$$ . Because $$\pi(J0)\,=\,\pi^{\prime}(J0)\,=\,J0$$ , we know both $$\pi$$ and $$\pi^{\prime}$$ must take $$J$$ -fixed-points to $$J$$ -fixed-points, i.e. $$\pi\Lambda_{1}\,=\,\gamma^{m}$$ and $$\pi^{\prime}\Lambda_{1}\,=\,\gamma^{m^{\prime}}$$ for some $$1\leq m,m^{\prime}\leq r$$ . It is easy to compute [25]</p> <p>From this we see $$m\,m^{\prime}\equiv\pm1$$ (mod $$\kappa$$ ), so $$^{\prime\prime}$$ is coprime to $$\kappa$$ . Hitting it with the Galois fusion-symmetry $$\pi\{m^{\prime}\}$$ , we see that we may assume $$\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$$ .</p> <p>Now use (4.2) to get $$\pi\gamma^{i}=\gamma^{i}$$ for all $$i$$ . Then $$\pi$$ equals the identity or $$\pi[1]$$ , depending on what $$\pi$$ does to $$\Lambda_{r}$$ .</p> <h1>4.4. The $$C$$ -series argument</h1> <p>By rank-level duality, we may take $$r\le k$$ . For now assume $$(r,k)\neq(2,3)$$ . Then we know $$\pi\Lambda_{1}=J^{a}\Lambda_{1}$$ and $$\pi\Lambda_{1}=J^{a^{\prime}}\Lambda_{1}$$ for some $$a,a^{\prime}$$ . Since $$\pi J0=\pi^{\prime}J0=J0$$ , (2.7b) says $$a=a^{\prime}=0$$ if $$k r$$ is odd. Since $$\chi_{\Lambda_{1}}[\Lambda_{1}]>0$$ (using (3.3)), $$S_{\Lambda_{1}\Lambda_{1}}=S_{J^{a}\Lambda_{1},J^{a^{\prime}}\Lambda_{1}}$$ implies that $$a=a^{\prime}$$ also holds when $$k r$$ is even, and hence we may assume (hitting with $$\pi[1]^{a}$$ ) that also</p>	[{"type": "text", "coordinates": [70, 93, 541, 122], "content": "$$k=1$$ is easy: $$P_{+}=\\{0,J0,\\Lambda_{r}\\}$$ and $$\\pi=i d$$ . is automatic. $$k=2$$ will be done later in\nthis subsection. Assume now that $$k\\geq3$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 123, 541, 266], "content": "From Proposition 4.1(b) we can write $$\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}$$ and $$\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}$$ . We know\n$$\\pi J0\\,=\\,J0$$ , so (2.7b) says $$\\pi$$ must take spinors to spinors, and nonspinors to nonspinors.\nThen we will have $$\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]$$ for any spinor $$\\psi$$ . Now if $$a^{\\prime}=1$$ , then $$\\pi$$ will\ntake the spinors which maximize $$\\chi_{\\Lambda_{1}}$$ , to those which minimize it. Both these maxima\nand minima can be easily found from (3.2). Thus we get that $$\\pi(S\\Lambda_{r})$$ equals $$k\\Lambda_{r}$$ (when\n$$k$$ odd) or $$S((k-1)\\Lambda_{r})$$ (when $$k$$ even). But the sets $$S\\Lambda_{r}$$ and $$k\\Lambda_{r}$$ have different cardi-\nnalities ( $$k\\Lambda_{r}$$ is a $$J$$ -fixed-point), and so can\u2019t get mapped to each other. Also, the fusions\n$$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{r}=\\Lambda_{r}$$ + $$(\\Lambda_{1}+\\Lambda_{r})$$ and $$J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)$$ + $$(J^{a+i+1}(k-$$\n$$1)\\Lambda_{r})$$ + $$J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})$$ have different numbers of weights on their right sides,\nso also $$\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 267, 541, 295], "content": "Thus $$a^{\\prime}=0$$ and $$\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}$$ for some $$b$$ . Similarly, $$a=0$$ . Hitting $$\\pi$$ with $$\\pi[1]^{b}$$ , we\nmay assume that $$\\pi$$ fixes $$\\Lambda_{r}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 296, 540, 326], "content": "Now assume $$\\pi$$ fixes $$\\Lambda_{\\ell}$$ , for $$1\\leq\\ell<r-1$$ . Then the fusion $$\\Lambda_{1}$$ \u00d7 $$\\Lambda_{\\ell}$$ says that $$\\pi\\Lambda_{\\ell+1}$$\nequals $$\\Lambda_{\\ell+1}$$ or $$\\Lambda_{1}+\\Lambda_{\\ell}$$ . But from (3.2) we find", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [124, 340, 527, 401], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 409, 391, 425], "content": "Hence $$\\pi$$ will fix $$\\Lambda_{\\ell+1}$$ if it fixes $$\\Lambda_{\\ell}$$ , concluding the argument.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [70, 428, 541, 501], "content": "Now consider the more interesting case: $$k=2$$ . Then $$\\kappa=2r+1$$ ; recall the weights\nin $$P_{+}(B_{r,2})$$ are the simple-currents $$0,J0$$ , the $$J$$ -fixed-points $$\\gamma^{1},\\ldots,\\gamma^{r}$$ (notation defined\nin \u00a73.2), and the spinors $$\\Lambda_{r},J\\Lambda_{r}$$ . Because $$\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0$$ , we know both $$\\pi$$ and\n$$\\pi^{\\prime}$$ must take $$J$$ -fixed-points to $$J$$ -fixed-points, i.e. $$\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}$$ and $$\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}$$ for some\n$$1\\leq m,m^{\\prime}\\leq r$$ . It is easy to compute [25]", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [247, 515, 365, 547], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [70, 560, 541, 589], "content": "From this we see $$m\\,m^{\\prime}\\equiv\\pm1$$ (mod $$\\kappa$$ ), so $$^{\\prime\\prime}$$ is coprime to $$\\kappa$$ . Hitting it with the Galois\nfusion-symmetry $$\\pi\\{m^{\\prime}\\}$$ , we see that we may assume $$\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [71, 590, 541, 619], "content": "Now use (4.2) to get $$\\pi\\gamma^{i}=\\gamma^{i}$$ for all $$i$$ . Then $$\\pi$$ equals the identity or $$\\pi[1]$$ , depending\non what $$\\pi$$ does to $$\\Lambda_{r}$$ .", "block_type": "text", "index": 10}, {"type": "title", "coordinates": [71, 634, 218, 649], "content": "4.4. The $$C$$ -series argument", "block_type": "title", "index": 11}, {"type": "text", "coordinates": [70, 657, 541, 715], "content": "By rank-level duality, we may take $$r\\le k$$ . For now assume $$(r,k)\\neq(2,3)$$ . Then we\nknow $$\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}$$ and $$\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}$$ for some $$a,a^{\\prime}$$ . Since $$\\pi J0=\\pi^{\\prime}J0=J0$$ , (2.7b) says\n$$a=a^{\\prime}=0$$ if $$k r$$ is odd. Since $$\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$$ (using (3.3)), $$S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}$$ implies that\n$$a=a^{\\prime}$$ also holds when $$k r$$ is even, and hence we may assume (hitting with $$\\pi[1]^{a}$$ ) that also", "block_type": "text", "index": 12}]	[{"type": "inline_equation", "coordinates": [95, 98, 124, 107], "content": "k=1", "score": 0.9, "index": 1}, {"type": "text", "coordinates": [124, 95, 171, 111], "content": " is easy: ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [171, 97, 259, 110], "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "score": 0.95, "index": 3}, {"type": "text", "coordinates": [259, 95, 286, 111], "content": " and ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [286, 98, 321, 107], "content": "\\pi=i d", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [321, 95, 400, 111], "content": ". is automatic. ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [401, 98, 430, 107], "content": "k=2", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [430, 95, 541, 111], "content": " will be done later in", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [72, 111, 252, 124], "content": "this subsection. Assume now that ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [252, 113, 281, 123], "content": "k\\geq3", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [281, 111, 285, 124], "content": ".", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [93, 125, 304, 140], "content": "From Proposition 4.1(b) we can write ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [304, 127, 372, 138], "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "score": 0.95, "index": 13}, {"type": "text", "coordinates": [373, 125, 402, 140], "content": " and ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [402, 124, 478, 138], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "score": 0.95, "index": 15}, {"type": "text", "coordinates": [478, 125, 541, 140], "content": ". We know", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [71, 142, 125, 151], "content": "\\pi J0\\,=\\,J0", "score": 0.88, "index": 17}, {"type": "text", "coordinates": [125, 140, 209, 155], "content": ", so (2.7b) says ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [209, 145, 217, 150], "content": "\\pi", "score": 0.81, "index": 19}, {"type": "text", "coordinates": [217, 140, 540, 155], "content": " must take spinors to spinors, and nonspinors to nonspinors.", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [70, 154, 172, 169], "content": "Then we will have ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [172, 153, 300, 168], "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [300, 154, 381, 169], "content": " for any spinor ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [382, 156, 390, 167], "content": "\\psi", "score": 0.9, "index": 24}, {"type": "text", "coordinates": [391, 154, 439, 169], "content": ". Now if ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [440, 155, 474, 165], "content": "a^{\\prime}=1", "score": 0.88, "index": 26}, {"type": "text", "coordinates": [474, 154, 508, 169], "content": ", then ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [509, 158, 516, 165], "content": "\\pi", "score": 0.61, "index": 28}, {"type": "text", "coordinates": [517, 154, 542, 169], "content": " will", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [70, 168, 251, 184], "content": "take the spinors which maximize ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [251, 173, 271, 182], "content": "\\chi_{\\Lambda_{1}}", "score": 0.86, "index": 31}, {"type": "text", "coordinates": [271, 168, 542, 184], "content": ", to those which minimize it. Both these maxima", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [70, 182, 403, 198], "content": "and minima can be easily found from (3.2). Thus we get that ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [403, 183, 442, 196], "content": "\\pi(S\\Lambda_{r})", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [442, 182, 482, 198], "content": " equals ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [482, 182, 503, 195], "content": "k\\Lambda_{r}", "score": 0.89, "index": 36}, {"type": "text", "coordinates": [504, 182, 541, 198], "content": " (when", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [70, 197, 79, 209], "content": "k", "score": 0.74, "index": 38}, {"type": "text", "coordinates": [79, 198, 126, 212], "content": " odd) or ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [127, 196, 196, 211], "content": "S((k-1)\\Lambda_{r})", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [196, 198, 236, 212], "content": " (when ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [236, 198, 244, 208], "content": "k", "score": 0.74, "index": 42}, {"type": "text", "coordinates": [244, 198, 357, 212], "content": " even). But the sets ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [357, 199, 379, 210], "content": "S\\Lambda_{r}", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [379, 198, 407, 212], "content": " and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [408, 198, 429, 210], "content": "k\\Lambda_{r}", "score": 0.89, "index": 46}, {"type": "text", "coordinates": [429, 198, 541, 212], "content": " have different cardi-", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [71, 212, 116, 226], "content": "nalities (", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [116, 211, 138, 224], "content": "k\\Lambda_{r}", "score": 0.88, "index": 49}, {"type": "text", "coordinates": [138, 212, 162, 226], "content": " is a ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [163, 212, 171, 223], "content": "J", "score": 0.83, "index": 51}, {"type": "text", "coordinates": [172, 212, 541, 226], "content": "-fixed-point), and so can\u2019t get mapped to each other. Also, the fusions", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [71, 225, 85, 238], "content": "\\Lambda_{1}", "score": 0.86, "index": 53}, {"type": "text", "coordinates": [86, 225, 101, 241], "content": " \u00d7", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [102, 225, 147, 238], "content": "\\Lambda_{r}=\\Lambda_{r}", "score": 0.89, "index": 55}, {"type": "text", "coordinates": [147, 225, 164, 241], "content": " + ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [164, 225, 215, 239], "content": "(\\Lambda_{1}+\\Lambda_{r})", "score": 0.89, "index": 57}, {"type": "text", "coordinates": [216, 225, 241, 241], "content": " and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [241, 226, 457, 240], "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "score": 0.85, "index": 59}, {"type": "text", "coordinates": [457, 225, 476, 241], "content": " + ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [477, 224, 541, 240], "content": "(J^{a+i+1}(k-", "score": 0.85, "index": 61}, {"type": "inline_equation", "coordinates": [70, 239, 99, 254], "content": "1)\\Lambda_{r})", "score": 0.85, "index": 62}, {"type": "text", "coordinates": [99, 237, 117, 257], "content": " + ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [118, 240, 254, 254], "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [254, 237, 541, 257], "content": " have different numbers of weights on their right sides,", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [70, 255, 109, 269], "content": "so also ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [110, 254, 204, 268], "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "score": 0.92, "index": 67}, {"type": "text", "coordinates": [204, 255, 209, 269], "content": ".", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [94, 268, 125, 285], "content": "Thus ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [126, 269, 160, 281], "content": "a^{\\prime}=0", "score": 0.9, "index": 70}, {"type": "text", "coordinates": [160, 268, 186, 285], "content": " and ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [186, 269, 251, 282], "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "score": 0.95, "index": 72}, {"type": "text", "coordinates": [252, 268, 304, 285], "content": " for some ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [304, 271, 310, 280], "content": "b", "score": 0.85, "index": 74}, {"type": "text", "coordinates": [310, 268, 372, 285], "content": ". Similarly, ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [372, 270, 403, 281], "content": "a=0", "score": 0.84, "index": 76}, {"type": "text", "coordinates": [403, 268, 453, 285], "content": ". Hitting ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [453, 272, 462, 280], "content": "\\pi", "score": 0.69, "index": 78}, {"type": "text", "coordinates": [462, 268, 492, 285], "content": " with ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [493, 268, 518, 283], "content": "\\pi[1]^{b}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [519, 268, 541, 285], "content": ", we", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [71, 284, 163, 298], "content": "may assume that", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [164, 286, 172, 295], "content": "\\pi", "score": 0.75, "index": 83}, {"type": "text", "coordinates": [173, 284, 202, 298], "content": " fixes ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [202, 285, 216, 296], "content": "\\Lambda_{r}", "score": 0.89, "index": 85}, {"type": "text", "coordinates": [217, 284, 220, 298], "content": ".", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [93, 297, 163, 315], "content": "Now assume ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [163, 301, 172, 309], "content": "\\pi", "score": 0.76, "index": 88}, {"type": "text", "coordinates": [172, 297, 201, 315], "content": " fixes ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [202, 300, 215, 311], "content": "\\Lambda_{\\ell}", "score": 0.88, "index": 90}, {"type": "text", "coordinates": [216, 297, 240, 315], "content": ", for ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [241, 300, 311, 311], "content": "1\\leq\\ell<r-1", "score": 0.91, "index": 92}, {"type": "text", "coordinates": [311, 297, 406, 315], "content": ". Then the fusion ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [406, 298, 421, 311], "content": "\\Lambda_{1}", "score": 0.79, "index": 94}, {"type": "text", "coordinates": [421, 297, 437, 315], "content": " \u00d7", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [438, 298, 452, 311], "content": "\\Lambda_{\\ell}", "score": 0.73, "index": 96}, {"type": "text", "coordinates": [452, 297, 506, 315], "content": "says that ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [506, 299, 539, 312], "content": "\\pi\\Lambda_{\\ell+1}", "score": 0.89, "index": 98}, {"type": "text", "coordinates": [71, 313, 106, 327], "content": "equals", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [107, 313, 133, 326], "content": "\\Lambda_{\\ell+1}", "score": 0.91, "index": 100}, {"type": "text", "coordinates": [133, 313, 150, 327], "content": " or ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [151, 313, 192, 325], "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "score": 0.91, "index": 102}, {"type": "text", "coordinates": [193, 313, 321, 327], "content": ". But from (3.2) we find", "score": 1.0, "index": 103}, {"type": "interline_equation", "coordinates": [124, 340, 527, 401], "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "score": 0.86, "index": 104}, {"type": "text", "coordinates": [70, 411, 106, 426], "content": "Hence ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [107, 417, 114, 423], "content": "\\pi", "score": 0.86, "index": 106}, {"type": "text", "coordinates": [114, 411, 157, 426], "content": " will fix ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [158, 413, 182, 425], "content": "\\Lambda_{\\ell+1}", "score": 0.92, "index": 108}, {"type": "text", "coordinates": [182, 411, 236, 426], "content": " if it fixes ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [236, 414, 249, 424], "content": "\\Lambda_{\\ell}", "score": 0.9, "index": 110}, {"type": "text", "coordinates": [250, 411, 390, 426], "content": ", concluding the argument.", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [94, 430, 311, 446], "content": "Now consider the more interesting case: ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [311, 432, 341, 441], "content": "k=2", "score": 0.91, "index": 113}, {"type": "text", "coordinates": [342, 430, 382, 446], "content": ". Then ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [382, 433, 440, 442], "content": "\\kappa=2r+1", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [440, 430, 540, 446], "content": "; recall the weights", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [69, 443, 85, 460], "content": "in ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [86, 446, 132, 459], "content": "P_{+}(B_{r,2})", "score": 0.94, "index": 118}, {"type": "text", "coordinates": [132, 443, 262, 460], "content": " are the simple-currents ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [262, 447, 288, 458], "content": "0,J0", "score": 0.92, "index": 120}, {"type": "text", "coordinates": [288, 443, 316, 460], "content": ", the ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [316, 447, 325, 456], "content": "J", "score": 0.88, "index": 122}, {"type": "text", "coordinates": [325, 443, 393, 460], "content": "-fixed-points ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [393, 445, 444, 458], "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "score": 0.92, "index": 124}, {"type": "text", "coordinates": [445, 443, 542, 460], "content": " (notation defined", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [69, 458, 206, 474], "content": "in \u00a73.2), and the spinors ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [207, 461, 247, 472], "content": "\\Lambda_{r},J\\Lambda_{r}", "score": 0.93, "index": 127}, {"type": "text", "coordinates": [248, 458, 304, 474], "content": ". Because ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [304, 460, 421, 473], "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "score": 0.93, "index": 129}, {"type": "text", "coordinates": [421, 458, 507, 474], "content": ", we know both ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [508, 464, 515, 470], "content": "\\pi", "score": 0.88, "index": 131}, {"type": "text", "coordinates": [516, 458, 541, 474], "content": " and", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [71, 475, 82, 484], "content": "\\pi^{\\prime}", "score": 0.9, "index": 133}, {"type": "text", "coordinates": [82, 472, 144, 489], "content": " must take ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [145, 475, 153, 484], "content": "J", "score": 0.9, "index": 135}, {"type": "text", "coordinates": [153, 472, 237, 489], "content": "-fixed-points to ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [238, 475, 246, 484], "content": "J", "score": 0.89, "index": 137}, {"type": "text", "coordinates": [246, 472, 339, 489], "content": "-fixed-points, i.e. ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [340, 475, 397, 487], "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [397, 472, 425, 489], "content": " and ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [425, 473, 489, 487], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "score": 0.94, "index": 141}, {"type": "text", "coordinates": [489, 472, 542, 489], "content": " for some", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [71, 489, 145, 501], "content": "1\\leq m,m^{\\prime}\\leq r", "score": 0.92, "index": 143}, {"type": "text", "coordinates": [145, 487, 287, 504], "content": ". It is easy to compute [25]", "score": 1.0, "index": 144}, {"type": "interline_equation", "coordinates": [247, 515, 365, 547], "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "score": 0.94, "index": 145}, {"type": "text", "coordinates": [70, 561, 165, 578], "content": "From this we see ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [165, 564, 225, 574], "content": "m\\,m^{\\prime}\\equiv\\pm1", "score": 0.8, "index": 147}, {"type": "text", "coordinates": [226, 561, 261, 578], "content": " (mod ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [262, 567, 269, 573], "content": "\\kappa", "score": 0.58, "index": 149}, {"type": "text", "coordinates": [269, 561, 296, 578], "content": "), so ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [297, 568, 307, 573], "content": "^{\\prime\\prime}", "score": 0.85, "index": 151}, {"type": "text", "coordinates": [308, 561, 385, 578], "content": " is coprime to ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [385, 566, 393, 573], "content": "\\kappa", "score": 0.74, "index": 153}, {"type": "text", "coordinates": [393, 561, 541, 578], "content": ". Hitting it with the Galois", "score": 1.0, "index": 154}, {"type": "text", "coordinates": [70, 577, 162, 592], "content": "fusion-symmetry ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [162, 578, 195, 591], "content": "\\pi\\{m^{\\prime}\\}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [196, 577, 349, 592], "content": ", we see that we may assume ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [350, 576, 441, 590], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "score": 0.92, "index": 158}, {"type": "text", "coordinates": [442, 577, 445, 592], "content": ".", "score": 1.0, "index": 159}, {"type": "text", "coordinates": [94, 591, 205, 607], "content": "Now use (4.2) to get ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [205, 591, 251, 605], "content": "\\pi\\gamma^{i}=\\gamma^{i}", "score": 0.93, "index": 161}, {"type": "text", "coordinates": [251, 591, 288, 607], "content": " for all ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [289, 594, 293, 603], "content": "i", "score": 0.75, "index": 163}, {"type": "text", "coordinates": [293, 591, 331, 607], "content": ". Then", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [332, 594, 340, 603], "content": "\\pi", "score": 0.72, "index": 165}, {"type": "text", "coordinates": [340, 591, 458, 607], "content": " equals the identity or ", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [459, 592, 479, 605], "content": "\\pi[1]", "score": 0.74, "index": 167}, {"type": "text", "coordinates": [479, 591, 540, 607], "content": ", depending", "score": 1.0, "index": 168}, {"type": "text", "coordinates": [70, 605, 117, 621], "content": "on what ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [118, 611, 125, 617], "content": "\\pi", "score": 0.87, "index": 170}, {"type": "text", "coordinates": [126, 605, 170, 621], "content": " does to ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [170, 608, 184, 619], "content": "\\Lambda_{r}", "score": 0.91, "index": 172}, {"type": "text", "coordinates": [185, 605, 189, 621], "content": ".", "score": 1.0, "index": 173}, {"type": "text", "coordinates": [71, 637, 119, 649], "content": "4.4. The ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [119, 639, 129, 647], "content": "C", "score": 0.87, "index": 175}, {"type": "text", "coordinates": [129, 637, 218, 649], "content": "-series argument", "score": 1.0, "index": 176}, {"type": "text", "coordinates": [94, 659, 284, 674], "content": "By rank-level duality, we may take ", "score": 1.0, "index": 177}, {"type": "inline_equation", "coordinates": [284, 661, 314, 672], "content": "r\\le k", "score": 0.93, "index": 178}, {"type": "text", "coordinates": [315, 659, 413, 674], "content": ". For now assume ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [413, 659, 484, 673], "content": "(r,k)\\neq(2,3)", "score": 0.91, "index": 180}, {"type": "text", "coordinates": [484, 659, 541, 674], "content": ". Then we", "score": 1.0, "index": 181}, {"type": "text", "coordinates": [70, 673, 102, 688], "content": "know ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [102, 676, 168, 686], "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "score": 0.92, "index": 183}, {"type": "text", "coordinates": [168, 673, 195, 688], "content": " and ", "score": 1.0, "index": 184}, {"type": "inline_equation", "coordinates": [196, 672, 265, 686], "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "score": 0.93, "index": 185}, {"type": "text", "coordinates": [265, 673, 318, 688], "content": " for some ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [318, 675, 340, 687], "content": "a,a^{\\prime}", "score": 0.91, "index": 187}, {"type": "text", "coordinates": [340, 673, 380, 688], "content": ". Since ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [380, 674, 475, 686], "content": "\\pi J0=\\pi^{\\prime}J0=J0", "score": 0.86, "index": 189}, {"type": "text", "coordinates": [475, 673, 540, 688], "content": ", (2.7b) says", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [71, 689, 126, 699], "content": "a=a^{\\prime}=0", "score": 0.92, "index": 191}, {"type": "text", "coordinates": [126, 685, 140, 707], "content": " if ", "score": 1.0, "index": 192}, {"type": "inline_equation", "coordinates": [140, 690, 153, 699], "content": "k r", "score": 0.89, "index": 193}, {"type": "text", "coordinates": [154, 685, 228, 707], "content": " is odd. Since ", "score": 1.0, "index": 194}, {"type": "inline_equation", "coordinates": [228, 689, 291, 702], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "score": 0.95, "index": 195}, {"type": "text", "coordinates": [291, 685, 366, 707], "content": " (using (3.3)), ", "score": 1.0, "index": 196}, {"type": "inline_equation", "coordinates": [367, 689, 473, 703], "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "score": 0.9, "index": 197}, {"type": "text", "coordinates": [473, 685, 542, 707], "content": " implies that", "score": 1.0, "index": 198}, {"type": "inline_equation", "coordinates": [71, 704, 103, 713], "content": "a=a^{\\prime}", "score": 0.92, "index": 199}, {"type": "text", "coordinates": [103, 702, 192, 717], "content": " also holds when ", "score": 1.0, "index": 200}, {"type": "inline_equation", "coordinates": [192, 704, 205, 713], "content": "k r", "score": 0.88, "index": 201}, {"type": "text", "coordinates": [205, 702, 459, 717], "content": " is even, and hence we may assume (hitting with ", "score": 1.0, "index": 202}, {"type": "inline_equation", "coordinates": [459, 703, 487, 716], "content": "\\pi[1]^{a}", "score": 0.84, "index": 203}, {"type": "text", "coordinates": [487, 702, 540, 717], "content": ") that also", "score": 1.0, "index": 204}]	[]	[{"type": "block", "coordinates": [124, 340, 527, 401], "content": "", "caption": ""}, {"type": "block", "coordinates": [247, 515, 365, 547], "content": "", "caption": ""}, {"type": "inline", "coordinates": [95, 98, 124, 107], "content": "k=1", "caption": ""}, {"type": "inline", "coordinates": [171, 97, 259, 110], "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "caption": ""}, {"type": "inline", "coordinates": [286, 98, 321, 107], "content": "\\pi=i d", "caption": ""}, {"type": "inline", "coordinates": [401, 98, 430, 107], "content": "k=2", "caption": ""}, {"type": "inline", "coordinates": [252, 113, 281, 123], "content": "k\\geq3", "caption": ""}, {"type": "inline", 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"inline", "coordinates": [102, 225, 147, 238], "content": "\\Lambda_{r}=\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [164, 225, 215, 239], "content": "(\\Lambda_{1}+\\Lambda_{r})", "caption": ""}, {"type": "inline", "coordinates": [241, 226, 457, 240], "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "caption": ""}, {"type": "inline", "coordinates": [477, 224, 541, 240], "content": "(J^{a+i+1}(k-", "caption": ""}, {"type": "inline", "coordinates": [70, 239, 99, 254], "content": "1)\\Lambda_{r})", "caption": ""}, {"type": "inline", "coordinates": [118, 240, 254, 254], "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "caption": ""}, {"type": "inline", "coordinates": [110, 254, 204, 268], "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [126, 269, 160, 281], "content": "a^{\\prime}=0", "caption": ""}, {"type": "inline", "coordinates": [186, 269, 251, 282], "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [304, 271, 310, 280], "content": "b", "caption": ""}, {"type": "inline", "coordinates": [372, 270, 403, 281], "content": "a=0", "caption": ""}, {"type": "inline", "coordinates": [453, 272, 462, 280], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [493, 268, 518, 283], "content": "\\pi[1]^{b}", "caption": ""}, {"type": "inline", "coordinates": [164, 286, 172, 295], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [202, 285, 216, 296], "content": "\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [163, 301, 172, 309], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [202, 300, 215, 311], "content": "\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [241, 300, 311, 311], "content": "1\\leq\\ell<r-1", "caption": ""}, {"type": "inline", "coordinates": [406, 298, 421, 311], "content": "\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [438, 298, 452, 311], "content": "\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [506, 299, 539, 312], "content": "\\pi\\Lambda_{\\ell+1}", "caption": ""}, {"type": "inline", "coordinates": [107, 313, 133, 326], "content": "\\Lambda_{\\ell+1}", "caption": ""}, {"type": "inline", "coordinates": [151, 313, 192, 325], "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [107, 417, 114, 423], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [158, 413, 182, 425], "content": "\\Lambda_{\\ell+1}", "caption": ""}, {"type": "inline", "coordinates": [236, 414, 249, 424], "content": "\\Lambda_{\\ell}", "caption": ""}, {"type": "inline", "coordinates": [311, 432, 341, 441], "content": "k=2", "caption": ""}, {"type": "inline", "coordinates": [382, 433, 440, 442], "content": "\\kappa=2r+1", "caption": ""}, {"type": "inline", "coordinates": [86, 446, 132, 459], "content": "P_{+}(B_{r,2})", "caption": ""}, {"type": "inline", "coordinates": [262, 447, 288, 458], "content": "0,J0", "caption": ""}, {"type": "inline", "coordinates": [316, 447, 325, 456], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [393, 445, 444, 458], "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "caption": ""}, {"type": "inline", "coordinates": [207, 461, 247, 472], "content": "\\Lambda_{r},J\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [304, 460, 421, 473], "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "caption": ""}, {"type": "inline", "coordinates": [508, 464, 515, 470], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [71, 475, 82, 484], "content": "\\pi^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [145, 475, 153, 484], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [238, 475, 246, 484], "content": "J", "caption": ""}, {"type": "inline", "coordinates": [340, 475, 397, 487], "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "caption": ""}, {"type": "inline", "coordinates": [425, 473, 489, 487], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "caption": ""}, {"type": "inline", "coordinates": [71, 489, 145, 501], "content": "1\\leq m,m^{\\prime}\\leq r", "caption": ""}, {"type": "inline", "coordinates": [165, 564, 225, 574], "content": "m\\,m^{\\prime}\\equiv\\pm1", "caption": ""}, {"type": "inline", "coordinates": [262, 567, 269, 573], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [297, 568, 307, 573], "content": "^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [385, 566, 393, 573], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [162, 578, 195, 591], "content": "\\pi\\{m^{\\prime}\\}", "caption": ""}, {"type": "inline", "coordinates": [350, 576, 441, 590], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [205, 591, 251, 605], "content": "\\pi\\gamma^{i}=\\gamma^{i}", "caption": ""}, {"type": "inline", "coordinates": [289, 594, 293, 603], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [332, 594, 340, 603], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [459, 592, 479, 605], "content": "\\pi[1]", "caption": ""}, {"type": "inline", "coordinates": [118, 611, 125, 617], "content": "\\pi", "caption": ""}, {"type": "inline", "coordinates": [170, 608, 184, 619], "content": "\\Lambda_{r}", "caption": ""}, {"type": "inline", "coordinates": [119, 639, 129, 647], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [284, 661, 314, 672], "content": "r\\le k", "caption": ""}, {"type": "inline", "coordinates": [413, 659, 484, 673], "content": "(r,k)\\neq(2,3)", "caption": ""}, {"type": "inline", "coordinates": [102, 676, 168, 686], "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [196, 672, 265, 686], "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "caption": ""}, {"type": "inline", "coordinates": [318, 675, 340, 687], "content": "a,a^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [380, 674, 475, 686], "content": "\\pi J0=\\pi^{\\prime}J0=J0", "caption": ""}, {"type": "inline", "coordinates": [71, 689, 126, 699], "content": "a=a^{\\prime}=0", "caption": ""}, {"type": "inline", "coordinates": [140, 690, 153, 699], "content": "k r", "caption": ""}, {"type": "inline", "coordinates": [228, 689, 291, 702], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "caption": ""}, {"type": "inline", "coordinates": [367, 689, 473, 703], "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "caption": ""}, {"type": "inline", "coordinates": [71, 704, 103, 713], "content": "a=a^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [192, 704, 205, 713], "content": "k r", "caption": ""}, {"type": "inline", "coordinates": [459, 703, 487, 716], "content": "\\pi[1]^{a}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "$k=1$ is easy: $P_{+}=\\{0,J0,\\Lambda_{r}\\}$ and $\\pi=i d$ . is automatic. $k=2$ will be done later in this subsection. Assume now that $k\\geq3$ . ", "page_idx": 17}, {"type": "text", "text": "From Proposition 4.1(b) we can write $\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}$ . We know $\\pi J0\\,=\\,J0$ , so (2.7b) says $\\pi$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]$ for any spinor $\\psi$ . Now if $a^{\\prime}=1$ , then $\\pi$ will take the spinors which maximize $\\chi_{\\Lambda_{1}}$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $\\pi(S\\Lambda_{r})$ equals $k\\Lambda_{r}$ (when $k$ odd) or $S((k-1)\\Lambda_{r})$ (when $k$ even). But the sets $S\\Lambda_{r}$ and $k\\Lambda_{r}$ have different cardinalities ( $k\\Lambda_{r}$ is a $J$ -fixed-point), and so can\u2019t get mapped to each other. Also, the fusions $\\Lambda_{1}$ \u00d7 $\\Lambda_{r}=\\Lambda_{r}$ + $(\\Lambda_{1}+\\Lambda_{r})$ and $J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)$ + $(J^{a+i+1}(k-$ $1)\\Lambda_{r})$ + $J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})$ have different numbers of weights on their right sides, so also $\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "Thus $a^{\\prime}=0$ and $\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}$ for some $b$ . Similarly, $a=0$ . Hitting $\\pi$ with $\\pi[1]^{b}$ , we may assume that $\\pi$ fixes $\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "Now assume $\\pi$ fixes $\\Lambda_{\\ell}$ , for $1\\leq\\ell<r-1$ . Then the fusion $\\Lambda_{1}$ \u00d7 $\\Lambda_{\\ell}$ says that $\\pi\\Lambda_{\\ell+1}$ equals $\\Lambda_{\\ell+1}$ or $\\Lambda_{1}+\\Lambda_{\\ell}$ . But from (3.2) we find ", "page_idx": 17}, {"type": "equation", "text": "$$\n\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}\n$$", "text_format": "latex", "page_idx": 17}, {"type": "text", "text": "Hence $\\pi$ will fix $\\Lambda_{\\ell+1}$ if it fixes $\\Lambda_{\\ell}$ , concluding the argument. ", "page_idx": 17}, {"type": "text", "text": "Now consider the more interesting case: $k=2$ . Then $\\kappa=2r+1$ ; recall the weights in $P_{+}(B_{r,2})$ are the simple-currents $0,J0$ , the $J$ -fixed-points $\\gamma^{1},\\ldots,\\gamma^{r}$ (notation defined in \u00a73.2), and the spinors $\\Lambda_{r},J\\Lambda_{r}$ . Because $\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0$ , we know both $\\pi$ and $\\pi^{\\prime}$ must take $J$ -fixed-points to $J$ -fixed-points, i.e. $\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}$ for some $1\\leq m,m^{\\prime}\\leq r$ . It is easy to compute [25] ", "page_idx": 17}, {"type": "equation", "text": "$$\n\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .\n$$", "text_format": "latex", "page_idx": 17}, {"type": "text", "text": "From this we see $m\\,m^{\\prime}\\equiv\\pm1$ (mod $\\kappa$ ), so $^{\\prime\\prime}$ is coprime to $\\kappa$ . Hitting it with the Galois fusion-symmetry $\\pi\\{m^{\\prime}\\}$ , we see that we may assume $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 17}, {"type": "text", "text": "Now use (4.2) to get $\\pi\\gamma^{i}=\\gamma^{i}$ for all $i$ . Then $\\pi$ equals the identity or $\\pi[1]$ , depending on what $\\pi$ does to $\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "4.4. The $C$ -series argument ", "text_level": 1, "page_idx": 17}, {"type": "text", "text": "By rank-level duality, we may take $r\\le k$ . For now assume $(r,k)\\neq(2,3)$ . Then we know $\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}$ and $\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}$ for some $a,a^{\\prime}$ . Since $\\pi J0=\\pi^{\\prime}J0=J0$ , (2.7b) says $a=a^{\\prime}=0$ if $k r$ is odd. Since $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$ (using (3.3)), $S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}$ implies that $a=a^{\\prime}$ also holds when $k r$ is even, and hence we may assume (hitting with $\\pi[1]^{a}$ ) that also $a=a^{\\prime}=0$ holds for $k r$ even. 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", "type": "text"}, {"bbox": [401, 98, 430, 107], "score": 0.92, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [430, 95, 541, 111], "score": 1.0, "content": " will be done later in", "type": "text"}], "index": 0}, {"bbox": [72, 111, 285, 124], "spans": [{"bbox": [72, 111, 252, 124], "score": 1.0, "content": "this subsection. Assume now that ", "type": "text"}, {"bbox": [252, 113, 281, 123], "score": 0.92, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 111, 285, 124], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [70, 123, 541, 266], "lines": [{"bbox": [93, 124, 541, 140], "spans": [{"bbox": [93, 125, 304, 140], "score": 1.0, "content": "From Proposition 4.1(b) we can write ", "type": "text"}, {"bbox": [304, 127, 372, 138], "score": 0.95, "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [373, 125, 402, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [402, 124, 478, 138], "score": 0.95, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 76}, {"bbox": [478, 125, 541, 140], "score": 1.0, "content": ". We know", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 155], "spans": [{"bbox": [71, 142, 125, 151], "score": 0.88, "content": "\\pi J0\\,=\\,J0", "type": "inline_equation", "height": 9, "width": 54}, {"bbox": [125, 140, 209, 155], "score": 1.0, "content": ", so (2.7b) says ", "type": "text"}, {"bbox": [209, 145, 217, 150], "score": 0.81, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [217, 140, 540, 155], "score": 1.0, "content": " must take spinors to spinors, and nonspinors to nonspinors.", "type": "text"}], "index": 3}, {"bbox": [70, 153, 542, 169], "spans": [{"bbox": [70, 154, 172, 169], "score": 1.0, "content": "Then we will have ", "type": "text"}, {"bbox": [172, 153, 300, 168], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [300, 154, 381, 169], "score": 1.0, "content": " for any spinor ", "type": "text"}, {"bbox": [382, 156, 390, 167], "score": 0.9, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [391, 154, 439, 169], "score": 1.0, "content": ". Now if ", "type": "text"}, {"bbox": [440, 155, 474, 165], "score": 0.88, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [474, 154, 508, 169], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [509, 158, 516, 165], "score": 0.61, "content": "\\pi", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [517, 154, 542, 169], "score": 1.0, "content": " will", "type": "text"}], "index": 4}, {"bbox": [70, 168, 542, 184], "spans": [{"bbox": [70, 168, 251, 184], "score": 1.0, "content": "take the spinors which maximize ", "type": "text"}, {"bbox": [251, 173, 271, 182], "score": 0.86, "content": "\\chi_{\\Lambda_{1}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [271, 168, 542, 184], "score": 1.0, "content": ", to those which minimize it. Both these maxima", "type": "text"}], "index": 5}, {"bbox": [70, 182, 541, 198], "spans": [{"bbox": [70, 182, 403, 198], "score": 1.0, "content": "and minima can be easily found from (3.2). Thus we get that ", "type": "text"}, {"bbox": [403, 183, 442, 196], "score": 0.92, "content": "\\pi(S\\Lambda_{r})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [442, 182, 482, 198], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [482, 182, 503, 195], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [504, 182, 541, 198], "score": 1.0, "content": " (when", "type": "text"}], "index": 6}, {"bbox": [70, 196, 541, 212], "spans": [{"bbox": [70, 197, 79, 209], "score": 0.74, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [79, 198, 126, 212], "score": 1.0, "content": " odd) or ", "type": "text"}, {"bbox": [127, 196, 196, 211], "score": 0.92, "content": "S((k-1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 69}, {"bbox": [196, 198, 236, 212], "score": 1.0, "content": " (when ", "type": "text"}, {"bbox": [236, 198, 244, 208], "score": 0.74, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [244, 198, 357, 212], "score": 1.0, "content": " even). But the sets ", "type": "text"}, {"bbox": [357, 199, 379, 210], "score": 0.92, "content": "S\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [379, 198, 407, 212], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 198, 429, 210], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [429, 198, 541, 212], "score": 1.0, "content": " have different cardi-", "type": "text"}], "index": 7}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 212, 116, 226], "score": 1.0, "content": "nalities (", "type": "text"}, {"bbox": [116, 211, 138, 224], "score": 0.88, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [138, 212, 162, 226], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [163, 212, 171, 223], "score": 0.83, "content": "J", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [172, 212, 541, 226], "score": 1.0, "content": "-fixed-point), and so can\u2019t get mapped to each other. Also, the fusions", "type": "text"}], "index": 8}, {"bbox": [71, 224, 541, 241], "spans": [{"bbox": [71, 225, 85, 238], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [86, 225, 101, 241], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [102, 225, 147, 238], "score": 0.89, "content": "\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [147, 225, 164, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [164, 225, 215, 239], "score": 0.89, "content": "(\\Lambda_{1}+\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [216, 225, 241, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 226, 457, 240], "score": 0.85, "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 216}, {"bbox": [457, 225, 476, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [477, 224, 541, 240], "score": 0.85, "content": "(J^{a+i+1}(k-", "type": "inline_equation", "height": 16, "width": 64}], "index": 9}, {"bbox": [70, 237, 541, 257], "spans": [{"bbox": [70, 239, 99, 254], "score": 0.85, "content": "1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 29}, {"bbox": [99, 237, 117, 257], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [118, 240, 254, 254], "score": 0.9, "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [254, 237, 541, 257], "score": 1.0, "content": " have different numbers of weights on their right sides,", "type": "text"}], "index": 10}, {"bbox": [70, 254, 209, 269], "spans": [{"bbox": [70, 255, 109, 269], "score": 1.0, "content": "so also ", "type": "text"}, {"bbox": [110, 254, 204, 268], "score": 0.92, "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [204, 255, 209, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 6.5}, {"type": "text", "bbox": [70, 267, 541, 295], "lines": [{"bbox": [94, 268, 541, 285], "spans": [{"bbox": [94, 268, 125, 285], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [126, 269, 160, 281], "score": 0.9, "content": "a^{\\prime}=0", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 268, 186, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 269, 251, 282], "score": 0.95, "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [252, 268, 304, 285], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [304, 271, 310, 280], "score": 0.85, "content": "b", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [310, 268, 372, 285], "score": 1.0, "content": ". Similarly, ", "type": "text"}, {"bbox": [372, 270, 403, 281], "score": 0.84, "content": "a=0", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [403, 268, 453, 285], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [453, 272, 462, 280], "score": 0.69, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [462, 268, 492, 285], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [493, 268, 518, 283], "score": 0.92, "content": "\\pi[1]^{b}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [519, 268, 541, 285], "score": 1.0, "content": ", we", "type": "text"}], "index": 12}, {"bbox": [71, 284, 220, 298], "spans": [{"bbox": [71, 284, 163, 298], "score": 1.0, "content": "may assume that", "type": "text"}, {"bbox": [164, 286, 172, 295], "score": 0.75, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [173, 284, 202, 298], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 285, 216, 296], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [217, 284, 220, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [70, 296, 540, 326], "lines": [{"bbox": [93, 297, 539, 315], "spans": [{"bbox": [93, 297, 163, 315], "score": 1.0, "content": "Now assume ", "type": "text"}, {"bbox": [163, 301, 172, 309], "score": 0.76, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [172, 297, 201, 315], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 300, 215, 311], "score": 0.88, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [216, 297, 240, 315], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [241, 300, 311, 311], "score": 0.91, "content": "1\\leq\\ell<r-1", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [311, 297, 406, 315], "score": 1.0, "content": ". Then the fusion ", "type": "text"}, {"bbox": [406, 298, 421, 311], "score": 0.79, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [421, 297, 437, 315], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [438, 298, 452, 311], "score": 0.73, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [452, 297, 506, 315], "score": 1.0, "content": "says that ", "type": "text"}, {"bbox": [506, 299, 539, 312], "score": 0.89, "content": "\\pi\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 33}], "index": 14}, {"bbox": [71, 313, 321, 327], "spans": [{"bbox": [71, 313, 106, 327], "score": 1.0, "content": "equals", "type": "text"}, {"bbox": [107, 313, 133, 326], "score": 0.91, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [133, 313, 150, 327], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [151, 313, 192, 325], "score": 0.91, "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [193, 313, 321, 327], "score": 1.0, "content": ". But from (3.2) we find", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [70, 409, 391, 425], "lines": [{"bbox": [70, 411, 390, 426], "spans": [{"bbox": [70, 411, 106, 426], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [107, 417, 114, 423], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [114, 411, 157, 426], "score": 1.0, "content": " will fix ", "type": "text"}, {"bbox": [158, 413, 182, 425], "score": 0.92, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [182, 411, 236, 426], "score": 1.0, "content": " if it fixes ", "type": "text"}, {"bbox": [236, 414, 249, 424], "score": 0.9, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [250, 411, 390, 426], "score": 1.0, "content": ", concluding the argument.", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 428, 541, 501], "lines": [{"bbox": [94, 430, 540, 446], "spans": [{"bbox": [94, 430, 311, 446], "score": 1.0, "content": "Now consider the more interesting case: ", "type": "text"}, {"bbox": [311, 432, 341, 441], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [342, 430, 382, 446], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [382, 433, 440, 442], "score": 0.91, "content": "\\kappa=2r+1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [440, 430, 540, 446], "score": 1.0, "content": "; recall the weights", "type": "text"}], "index": 18}, {"bbox": [69, 443, 542, 460], "spans": [{"bbox": [69, 443, 85, 460], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [86, 446, 132, 459], "score": 0.94, "content": "P_{+}(B_{r,2})", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [132, 443, 262, 460], "score": 1.0, "content": " are the simple-currents ", "type": "text"}, {"bbox": [262, 447, 288, 458], "score": 0.92, "content": "0,J0", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [288, 443, 316, 460], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [316, 447, 325, 456], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [325, 443, 393, 460], "score": 1.0, "content": "-fixed-points ", "type": "text"}, {"bbox": [393, 445, 444, 458], "score": 0.92, "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [445, 443, 542, 460], "score": 1.0, "content": " (notation defined", "type": "text"}], "index": 19}, {"bbox": [69, 458, 541, 474], "spans": [{"bbox": [69, 458, 206, 474], "score": 1.0, "content": "in \u00a73.2), and the spinors ", "type": "text"}, {"bbox": [207, 461, 247, 472], "score": 0.93, "content": "\\Lambda_{r},J\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [248, 458, 304, 474], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [304, 460, 421, 473], "score": 0.93, "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [421, 458, 507, 474], "score": 1.0, "content": ", we know both ", "type": "text"}, {"bbox": [508, 464, 515, 470], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 458, 541, 474], "score": 1.0, "content": " and", "type": "text"}], "index": 20}, {"bbox": [71, 472, 542, 489], "spans": [{"bbox": [71, 475, 82, 484], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [82, 472, 144, 489], "score": 1.0, "content": " must take ", "type": "text"}, {"bbox": [145, 475, 153, 484], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [153, 472, 237, 489], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [238, 475, 246, 484], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [246, 472, 339, 489], "score": 1.0, "content": "-fixed-points, i.e. ", "type": "text"}, {"bbox": [340, 475, 397, 487], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [397, 472, 425, 489], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [425, 473, 489, 487], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [489, 472, 542, 489], "score": 1.0, "content": " for some", "type": "text"}], "index": 21}, {"bbox": [71, 487, 287, 504], "spans": [{"bbox": [71, 489, 145, 501], "score": 0.92, "content": "1\\leq m,m^{\\prime}\\leq r", "type": "inline_equation", "height": 12, "width": 74}, {"bbox": [145, 487, 287, 504], "score": 1.0, "content": ". It is easy to compute [25]", "type": "text"}], "index": 22}], "index": 20}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [70, 560, 541, 589], "lines": [{"bbox": [70, 561, 541, 578], "spans": [{"bbox": [70, 561, 165, 578], "score": 1.0, "content": "From this we see ", "type": "text"}, {"bbox": [165, 564, 225, 574], "score": 0.8, "content": "m\\,m^{\\prime}\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [226, 561, 261, 578], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [262, 567, 269, 573], "score": 0.58, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [269, 561, 296, 578], "score": 1.0, "content": "), so ", "type": "text"}, {"bbox": [297, 568, 307, 573], "score": 0.85, "content": "^{\\prime\\prime}", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [308, 561, 385, 578], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [385, 566, 393, 573], "score": 0.74, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [393, 561, 541, 578], "score": 1.0, "content": ". Hitting it with the Galois", "type": "text"}], "index": 24}, {"bbox": [70, 576, 445, 592], "spans": [{"bbox": [70, 577, 162, 592], "score": 1.0, "content": "fusion-symmetry ", "type": "text"}, {"bbox": [162, 578, 195, 591], "score": 0.93, "content": "\\pi\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [196, 577, 349, 592], "score": 1.0, "content": ", we see that we may assume ", "type": "text"}, {"bbox": [350, 576, 441, 590], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [442, 577, 445, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [71, 590, 541, 619], "lines": [{"bbox": [94, 591, 540, 607], "spans": [{"bbox": [94, 591, 205, 607], "score": 1.0, "content": "Now use (4.2) to get ", "type": "text"}, {"bbox": [205, 591, 251, 605], "score": 0.93, "content": "\\pi\\gamma^{i}=\\gamma^{i}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [251, 591, 288, 607], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 594, 293, 603], "score": 0.75, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [293, 591, 331, 607], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [332, 594, 340, 603], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [340, 591, 458, 607], "score": 1.0, "content": " equals the identity or ", "type": "text"}, {"bbox": [459, 592, 479, 605], "score": 0.74, "content": "\\pi[1]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [479, 591, 540, 607], "score": 1.0, "content": ", depending", "type": "text"}], "index": 26}, {"bbox": [70, 605, 189, 621], "spans": [{"bbox": [70, 605, 117, 621], "score": 1.0, "content": "on what ", "type": "text"}, {"bbox": [118, 611, 125, 617], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [126, 605, 170, 621], "score": 1.0, "content": " does to ", "type": "text"}, {"bbox": [170, 608, 184, 619], "score": 0.91, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [185, 605, 189, 621], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "title", "bbox": [71, 634, 218, 649], "lines": [{"bbox": [71, 637, 218, 649], "spans": [{"bbox": [71, 637, 119, 649], "score": 1.0, "content": "4.4. The ", "type": "text"}, {"bbox": [119, 639, 129, 647], "score": 0.87, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [129, 637, 218, 649], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [70, 657, 541, 715], "lines": [{"bbox": [94, 659, 541, 674], "spans": [{"bbox": [94, 659, 284, 674], "score": 1.0, "content": "By rank-level duality, we may take ", "type": "text"}, {"bbox": [284, 661, 314, 672], "score": 0.93, "content": "r\\le k", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [315, 659, 413, 674], "score": 1.0, "content": ". For now assume ", "type": "text"}, {"bbox": [413, 659, 484, 673], "score": 0.91, "content": "(r,k)\\neq(2,3)", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [484, 659, 541, 674], "score": 1.0, "content": ". Then we", "type": "text"}], "index": 29}, {"bbox": [70, 672, 540, 688], "spans": [{"bbox": [70, 673, 102, 688], "score": 1.0, "content": "know ", "type": "text"}, {"bbox": [102, 676, 168, 686], "score": 0.92, "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 66}, {"bbox": [168, 673, 195, 688], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [196, 672, 265, 686], "score": 0.93, "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [265, 673, 318, 688], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [318, 675, 340, 687], "score": 0.91, "content": "a,a^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [340, 673, 380, 688], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [380, 674, 475, 686], "score": 0.86, "content": "\\pi J0=\\pi^{\\prime}J0=J0", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [475, 673, 540, 688], "score": 1.0, "content": ", (2.7b) says", "type": "text"}], "index": 30}, {"bbox": [71, 685, 542, 707], "spans": [{"bbox": [71, 689, 126, 699], "score": 0.92, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 55}, {"bbox": [126, 685, 140, 707], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [140, 690, 153, 699], "score": 0.89, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [154, 685, 228, 707], "score": 1.0, "content": " is odd. Since ", "type": "text"}, {"bbox": [228, 689, 291, 702], "score": 0.95, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [291, 685, 366, 707], "score": 1.0, "content": " (using (3.3)), ", "type": "text"}, {"bbox": [367, 689, 473, 703], "score": 0.9, "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [473, 685, 542, 707], "score": 1.0, "content": " implies that", "type": "text"}], "index": 31}, {"bbox": [71, 702, 540, 717], "spans": [{"bbox": [71, 704, 103, 713], "score": 0.92, "content": "a=a^{\\prime}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [103, 702, 192, 717], "score": 1.0, "content": " also holds when ", "type": "text"}, {"bbox": [192, 704, 205, 713], "score": 0.88, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [205, 702, 459, 717], "score": 1.0, "content": " is even, and hence we may assume (hitting with ", "type": "text"}, {"bbox": [459, 703, 487, 716], "score": 0.84, "content": "\\pi[1]^{a}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [487, 702, 540, 717], "score": 1.0, "content": ") that also", "type": "text"}], "index": 32}], "index": 30.5}], "layout_bboxes": [], "page_idx": 17, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 741], "lines": [{"bbox": [298, 731, 313, 744], "spans": [{"bbox": [298, 731, 313, 744], "score": 1.0, "content": "18", "type": "text"}]}]}, {"type": "discarded", "bbox": [71, 71, 219, 85], "lines": [{"bbox": [72, 74, 219, 86], "spans": [{"bbox": [72, 74, 120, 86], "score": 1.0, "content": "4.3. The ", "type": "text"}, {"bbox": [120, 75, 131, 84], "score": 0.86, "content": "B", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [131, 74, 219, 86], "score": 1.0, "content": "-series argument", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 93, 541, 122], "lines": [{"bbox": [95, 95, 541, 111], "spans": [{"bbox": [95, 98, 124, 107], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 95, 171, 111], "score": 1.0, "content": " is easy: ", "type": "text"}, {"bbox": [171, 97, 259, 110], "score": 0.95, "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [259, 95, 286, 111], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 98, 321, 107], "score": 0.92, "content": "\\pi=i d", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [321, 95, 400, 111], "score": 1.0, "content": ". is automatic. ", "type": "text"}, {"bbox": [401, 98, 430, 107], "score": 0.92, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [430, 95, 541, 111], "score": 1.0, "content": " will be done later in", "type": "text"}], "index": 0}, {"bbox": [72, 111, 285, 124], "spans": [{"bbox": [72, 111, 252, 124], "score": 1.0, "content": "this subsection. Assume now that ", "type": "text"}, {"bbox": [252, 113, 281, 123], "score": 0.92, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 111, 285, 124], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [72, 95, 541, 124]}, {"type": "text", "bbox": [70, 123, 541, 266], "lines": [{"bbox": [93, 124, 541, 140], "spans": [{"bbox": [93, 125, 304, 140], "score": 1.0, "content": "From Proposition 4.1(b) we can write ", "type": "text"}, {"bbox": [304, 127, 372, 138], "score": 0.95, "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [373, 125, 402, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [402, 124, 478, 138], "score": 0.95, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 76}, {"bbox": [478, 125, 541, 140], "score": 1.0, "content": ". We know", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 155], "spans": [{"bbox": [71, 142, 125, 151], "score": 0.88, "content": "\\pi J0\\,=\\,J0", "type": "inline_equation", "height": 9, "width": 54}, {"bbox": [125, 140, 209, 155], "score": 1.0, "content": ", so (2.7b) says ", "type": "text"}, {"bbox": [209, 145, 217, 150], "score": 0.81, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [217, 140, 540, 155], "score": 1.0, "content": " must take spinors to spinors, and nonspinors to nonspinors.", "type": "text"}], "index": 3}, {"bbox": [70, 153, 542, 169], "spans": [{"bbox": [70, 154, 172, 169], "score": 1.0, "content": "Then we will have ", "type": "text"}, {"bbox": [172, 153, 300, 168], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [300, 154, 381, 169], "score": 1.0, "content": " for any spinor ", "type": "text"}, {"bbox": [382, 156, 390, 167], "score": 0.9, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [391, 154, 439, 169], "score": 1.0, "content": ". Now if ", "type": "text"}, {"bbox": [440, 155, 474, 165], "score": 0.88, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [474, 154, 508, 169], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [509, 158, 516, 165], "score": 0.61, "content": "\\pi", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [517, 154, 542, 169], "score": 1.0, "content": " will", "type": "text"}], "index": 4}, {"bbox": [70, 168, 542, 184], "spans": [{"bbox": [70, 168, 251, 184], "score": 1.0, "content": "take the spinors which maximize ", "type": "text"}, {"bbox": [251, 173, 271, 182], "score": 0.86, "content": "\\chi_{\\Lambda_{1}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [271, 168, 542, 184], "score": 1.0, "content": ", to those which minimize it. Both these maxima", "type": "text"}], "index": 5}, {"bbox": [70, 182, 541, 198], "spans": [{"bbox": [70, 182, 403, 198], "score": 1.0, "content": "and minima can be easily found from (3.2). Thus we get that ", "type": "text"}, {"bbox": [403, 183, 442, 196], "score": 0.92, "content": "\\pi(S\\Lambda_{r})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [442, 182, 482, 198], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [482, 182, 503, 195], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [504, 182, 541, 198], "score": 1.0, "content": " (when", "type": "text"}], "index": 6}, {"bbox": [70, 196, 541, 212], "spans": [{"bbox": [70, 197, 79, 209], "score": 0.74, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [79, 198, 126, 212], "score": 1.0, "content": " odd) or ", "type": "text"}, {"bbox": [127, 196, 196, 211], "score": 0.92, "content": "S((k-1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 69}, {"bbox": [196, 198, 236, 212], "score": 1.0, "content": " (when ", "type": "text"}, {"bbox": [236, 198, 244, 208], "score": 0.74, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [244, 198, 357, 212], "score": 1.0, "content": " even). But the sets ", "type": "text"}, {"bbox": [357, 199, 379, 210], "score": 0.92, "content": "S\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [379, 198, 407, 212], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 198, 429, 210], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [429, 198, 541, 212], "score": 1.0, "content": " have different cardi-", "type": "text"}], "index": 7}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 212, 116, 226], "score": 1.0, "content": "nalities (", "type": "text"}, {"bbox": [116, 211, 138, 224], "score": 0.88, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [138, 212, 162, 226], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [163, 212, 171, 223], "score": 0.83, "content": "J", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [172, 212, 541, 226], "score": 1.0, "content": "-fixed-point), and so can\u2019t get mapped to each other. Also, the fusions", "type": "text"}], "index": 8}, {"bbox": [71, 224, 541, 241], "spans": [{"bbox": [71, 225, 85, 238], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [86, 225, 101, 241], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [102, 225, 147, 238], "score": 0.89, "content": "\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [147, 225, 164, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [164, 225, 215, 239], "score": 0.89, "content": "(\\Lambda_{1}+\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [216, 225, 241, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 226, 457, 240], "score": 0.85, "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 216}, {"bbox": [457, 225, 476, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [477, 224, 541, 240], "score": 0.85, "content": "(J^{a+i+1}(k-", "type": "inline_equation", "height": 16, "width": 64}], "index": 9}, {"bbox": [70, 237, 541, 257], "spans": [{"bbox": [70, 239, 99, 254], "score": 0.85, "content": "1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 29}, {"bbox": [99, 237, 117, 257], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [118, 240, 254, 254], "score": 0.9, "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [254, 237, 541, 257], "score": 1.0, "content": " have different numbers of weights on their right sides,", "type": "text"}], "index": 10}, {"bbox": [70, 254, 209, 269], "spans": [{"bbox": [70, 255, 109, 269], "score": 1.0, "content": "so also ", "type": "text"}, {"bbox": [110, 254, 204, 268], "score": 0.92, "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [204, 255, 209, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 6.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 124, 542, 269]}, {"type": "text", "bbox": [70, 267, 541, 295], "lines": [{"bbox": [94, 268, 541, 285], "spans": [{"bbox": [94, 268, 125, 285], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [126, 269, 160, 281], "score": 0.9, "content": "a^{\\prime}=0", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 268, 186, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 269, 251, 282], "score": 0.95, "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [252, 268, 304, 285], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [304, 271, 310, 280], "score": 0.85, "content": "b", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [310, 268, 372, 285], "score": 1.0, "content": ". Similarly, ", "type": "text"}, {"bbox": [372, 270, 403, 281], "score": 0.84, "content": "a=0", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [403, 268, 453, 285], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [453, 272, 462, 280], "score": 0.69, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [462, 268, 492, 285], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [493, 268, 518, 283], "score": 0.92, "content": "\\pi[1]^{b}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [519, 268, 541, 285], "score": 1.0, "content": ", we", "type": "text"}], "index": 12}, {"bbox": [71, 284, 220, 298], "spans": [{"bbox": [71, 284, 163, 298], "score": 1.0, "content": "may assume that", "type": "text"}, {"bbox": [164, 286, 172, 295], "score": 0.75, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [173, 284, 202, 298], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 285, 216, 296], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [217, 284, 220, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [71, 268, 541, 298]}, {"type": "text", "bbox": [70, 296, 540, 326], "lines": [{"bbox": [93, 297, 539, 315], "spans": [{"bbox": [93, 297, 163, 315], "score": 1.0, "content": "Now assume ", "type": "text"}, {"bbox": [163, 301, 172, 309], "score": 0.76, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [172, 297, 201, 315], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 300, 215, 311], "score": 0.88, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [216, 297, 240, 315], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [241, 300, 311, 311], "score": 0.91, "content": "1\\leq\\ell<r-1", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [311, 297, 406, 315], "score": 1.0, "content": ". Then the fusion ", "type": "text"}, {"bbox": [406, 298, 421, 311], "score": 0.79, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [421, 297, 437, 315], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [438, 298, 452, 311], "score": 0.73, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [452, 297, 506, 315], "score": 1.0, "content": "says that ", "type": "text"}, {"bbox": [506, 299, 539, 312], "score": 0.89, "content": "\\pi\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 33}], "index": 14}, {"bbox": [71, 313, 321, 327], "spans": [{"bbox": [71, 313, 106, 327], "score": 1.0, "content": "equals", "type": "text"}, {"bbox": [107, 313, 133, 326], "score": 0.91, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [133, 313, 150, 327], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [151, 313, 192, 325], "score": 0.91, "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [193, 313, 321, 327], "score": 1.0, "content": ". But from (3.2) we find", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [71, 297, 539, 327]}, {"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 409, 391, 425], "lines": [{"bbox": [70, 411, 390, 426], "spans": [{"bbox": [70, 411, 106, 426], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [107, 417, 114, 423], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [114, 411, 157, 426], "score": 1.0, "content": " will fix ", "type": "text"}, {"bbox": [158, 413, 182, 425], "score": 0.92, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [182, 411, 236, 426], "score": 1.0, "content": " if it fixes ", "type": "text"}, {"bbox": [236, 414, 249, 424], "score": 0.9, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [250, 411, 390, 426], "score": 1.0, "content": ", concluding the argument.", "type": "text"}], "index": 17}], "index": 17, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 411, 390, 426]}, {"type": "text", "bbox": [70, 428, 541, 501], "lines": [{"bbox": [94, 430, 540, 446], "spans": [{"bbox": [94, 430, 311, 446], "score": 1.0, "content": "Now consider the more interesting case: ", "type": "text"}, {"bbox": [311, 432, 341, 441], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [342, 430, 382, 446], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [382, 433, 440, 442], "score": 0.91, "content": "\\kappa=2r+1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [440, 430, 540, 446], "score": 1.0, "content": "; recall the weights", "type": "text"}], "index": 18}, {"bbox": [69, 443, 542, 460], "spans": [{"bbox": [69, 443, 85, 460], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [86, 446, 132, 459], "score": 0.94, "content": "P_{+}(B_{r,2})", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [132, 443, 262, 460], "score": 1.0, "content": " are the simple-currents ", "type": "text"}, {"bbox": [262, 447, 288, 458], "score": 0.92, "content": "0,J0", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [288, 443, 316, 460], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [316, 447, 325, 456], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [325, 443, 393, 460], "score": 1.0, "content": "-fixed-points ", "type": "text"}, {"bbox": [393, 445, 444, 458], "score": 0.92, "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [445, 443, 542, 460], "score": 1.0, "content": " (notation defined", "type": "text"}], "index": 19}, {"bbox": [69, 458, 541, 474], "spans": [{"bbox": [69, 458, 206, 474], "score": 1.0, "content": "in \u00a73.2), and the spinors ", "type": "text"}, {"bbox": [207, 461, 247, 472], "score": 0.93, "content": "\\Lambda_{r},J\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [248, 458, 304, 474], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [304, 460, 421, 473], "score": 0.93, "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [421, 458, 507, 474], "score": 1.0, "content": ", we know both ", "type": "text"}, {"bbox": [508, 464, 515, 470], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 458, 541, 474], "score": 1.0, "content": " and", "type": "text"}], "index": 20}, {"bbox": [71, 472, 542, 489], "spans": [{"bbox": [71, 475, 82, 484], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [82, 472, 144, 489], "score": 1.0, "content": " must take ", "type": "text"}, {"bbox": [145, 475, 153, 484], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [153, 472, 237, 489], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [238, 475, 246, 484], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [246, 472, 339, 489], "score": 1.0, "content": "-fixed-points, i.e. ", "type": "text"}, {"bbox": [340, 475, 397, 487], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [397, 472, 425, 489], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [425, 473, 489, 487], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [489, 472, 542, 489], "score": 1.0, "content": " for some", "type": "text"}], "index": 21}, {"bbox": [71, 487, 287, 504], "spans": [{"bbox": [71, 489, 145, 501], "score": 0.92, "content": "1\\leq m,m^{\\prime}\\leq r", "type": "inline_equation", "height": 12, "width": 74}, {"bbox": [145, 487, 287, 504], "score": 1.0, "content": ". It is easy to compute [25]", "type": "text"}], "index": 22}], "index": 20, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [69, 430, 542, 504]}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 560, 541, 589], "lines": [{"bbox": [70, 561, 541, 578], "spans": [{"bbox": [70, 561, 165, 578], "score": 1.0, "content": "From this we see ", "type": "text"}, {"bbox": [165, 564, 225, 574], "score": 0.8, "content": "m\\,m^{\\prime}\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [226, 561, 261, 578], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [262, 567, 269, 573], "score": 0.58, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [269, 561, 296, 578], "score": 1.0, "content": "), so ", "type": "text"}, {"bbox": [297, 568, 307, 573], "score": 0.85, "content": "^{\\prime\\prime}", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [308, 561, 385, 578], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [385, 566, 393, 573], "score": 0.74, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [393, 561, 541, 578], "score": 1.0, "content": ". Hitting it with the Galois", "type": "text"}], "index": 24}, {"bbox": [70, 576, 445, 592], "spans": [{"bbox": [70, 577, 162, 592], "score": 1.0, "content": "fusion-symmetry ", "type": "text"}, {"bbox": [162, 578, 195, 591], "score": 0.93, "content": "\\pi\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [196, 577, 349, 592], "score": 1.0, "content": ", we see that we may assume ", "type": "text"}, {"bbox": [350, 576, 441, 590], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [442, 577, 445, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 561, 541, 592]}, {"type": "text", "bbox": [71, 590, 541, 619], "lines": [{"bbox": [94, 591, 540, 607], "spans": [{"bbox": [94, 591, 205, 607], "score": 1.0, "content": "Now use (4.2) to get ", "type": "text"}, {"bbox": [205, 591, 251, 605], "score": 0.93, "content": "\\pi\\gamma^{i}=\\gamma^{i}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [251, 591, 288, 607], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 594, 293, 603], "score": 0.75, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [293, 591, 331, 607], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [332, 594, 340, 603], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [340, 591, 458, 607], "score": 1.0, "content": " equals the identity or ", "type": "text"}, {"bbox": [459, 592, 479, 605], "score": 0.74, "content": "\\pi[1]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [479, 591, 540, 607], "score": 1.0, "content": ", depending", "type": "text"}], "index": 26}, {"bbox": [70, 605, 189, 621], "spans": [{"bbox": [70, 605, 117, 621], "score": 1.0, "content": "on what ", "type": "text"}, {"bbox": [118, 611, 125, 617], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [126, 605, 170, 621], "score": 1.0, "content": " does to ", "type": "text"}, {"bbox": [170, 608, 184, 619], "score": 0.91, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [185, 605, 189, 621], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 591, 540, 621]}, {"type": "title", "bbox": [71, 634, 218, 649], "lines": [{"bbox": [71, 637, 218, 649], "spans": [{"bbox": [71, 637, 119, 649], "score": 1.0, "content": "4.4. The ", "type": "text"}, {"bbox": [119, 639, 129, 647], "score": 0.87, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [129, 637, 218, 649], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 657, 541, 715], "lines": [{"bbox": [94, 659, 541, 674], "spans": [{"bbox": [94, 659, 284, 674], "score": 1.0, "content": "By rank-level duality, we may take ", "type": "text"}, {"bbox": [284, 661, 314, 672], "score": 0.93, "content": "r\\le k", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [315, 659, 413, 674], "score": 1.0, "content": ". For now assume ", "type": "text"}, {"bbox": [413, 659, 484, 673], "score": 0.91, "content": "(r,k)\\neq(2,3)", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [484, 659, 541, 674], "score": 1.0, "content": ". Then we", "type": "text"}], "index": 29}, {"bbox": [70, 672, 540, 688], "spans": [{"bbox": [70, 673, 102, 688], "score": 1.0, "content": "know ", "type": "text"}, {"bbox": [102, 676, 168, 686], "score": 0.92, "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 66}, {"bbox": [168, 673, 195, 688], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [196, 672, 265, 686], "score": 0.93, "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [265, 673, 318, 688], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [318, 675, 340, 687], "score": 0.91, "content": "a,a^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [340, 673, 380, 688], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [380, 674, 475, 686], "score": 0.86, "content": "\\pi J0=\\pi^{\\prime}J0=J0", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [475, 673, 540, 688], "score": 1.0, "content": ", (2.7b) says", "type": "text"}], "index": 30}, {"bbox": [71, 685, 542, 707], "spans": [{"bbox": [71, 689, 126, 699], "score": 0.92, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 55}, {"bbox": [126, 685, 140, 707], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [140, 690, 153, 699], "score": 0.89, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [154, 685, 228, 707], "score": 1.0, "content": " is odd. Since ", "type": "text"}, {"bbox": [228, 689, 291, 702], "score": 0.95, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [291, 685, 366, 707], "score": 1.0, "content": " (using (3.3)), ", "type": "text"}, {"bbox": [367, 689, 473, 703], "score": 0.9, "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [473, 685, 542, 707], "score": 1.0, "content": " implies that", "type": "text"}], "index": 31}, {"bbox": [71, 702, 540, 717], "spans": [{"bbox": [71, 704, 103, 713], "score": 0.92, "content": "a=a^{\\prime}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [103, 702, 192, 717], "score": 1.0, "content": " also holds when ", "type": "text"}, {"bbox": [192, 704, 205, 713], "score": 0.88, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [205, 702, 459, 717], "score": 1.0, "content": " is even, and hence we may assume (hitting with ", "type": "text"}, {"bbox": [459, 703, 487, 716], "score": 0.84, "content": "\\pi[1]^{a}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [487, 702, 540, 717], "score": 1.0, "content": ") that also", "type": "text"}], "index": 32}, {"bbox": [71, 72, 542, 90], "spans": [{"bbox": [71, 75, 126, 84], "score": 0.93, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 9, "width": 55, "cross_page": true}, {"bbox": [126, 72, 178, 90], "score": 1.0, "content": " holds for ", "type": "text", "cross_page": true}, {"bbox": [178, 75, 191, 84], "score": 0.39, "content": "k r", "type": "inline_equation", "height": 9, "width": 13, "cross_page": true}, {"bbox": [191, 72, 312, 90], "score": 1.0, "content": " even. From the fusion ", "type": "text", "cross_page": true}, {"bbox": [313, 75, 326, 86], "score": 0.64, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [327, 72, 344, 90], "score": 1.0, "content": " \u00d7 ", "type": "text", "cross_page": true}, {"bbox": [344, 74, 358, 86], "score": 0.63, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [358, 72, 398, 90], "score": 1.0, "content": "we get ", "type": "text", "cross_page": true}, {"bbox": [398, 75, 529, 87], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 12, "width": 131, "cross_page": true}, {"bbox": [529, 72, 542, 90], "score": 1.0, "content": " if", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [71, 89, 408, 102], "spans": [{"bbox": [71, 90, 120, 100], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 49, "cross_page": true}, {"bbox": [121, 89, 145, 102], "score": 1.0, "content": "; for ", "type": "text", "cross_page": true}, {"bbox": [146, 90, 174, 99], "score": 0.92, "content": "r<k", "type": "inline_equation", "height": 9, "width": 28, "cross_page": true}, {"bbox": [174, 89, 408, 102], "score": 1.0, "content": " conclude the argument with the calculation", "type": "text", "cross_page": true}], "index": 1}], "index": 30.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 659, 542, 717]}]}
0002044v1	23	28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products of $$E_{8}$$ representations, Alg., Groups Geom. 3 (1986), 286–328. 29. W. G. McKay, J. Patera and D. W. Rand, “Tables of representations of simple Lie algebras”, Vol. 1, Centre de Recherches Math´ematiques, Univ´ersit´e de Montr´eal, 1990. 30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in: “Interface between physics and mathematics”, World-Scientific, 1994, (W. Nahm and J.-M. Shen, Eds.), World-Scientific, Singapore, 1994. 31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents, Phys. Lett. B244 (1990), 255–260. 32. V. G. Turaev, “Quantum invariants of knots and 3-manifolds”, Walter de Gruyter, Berlin, 1994. 33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. 300 (1988), 360–376. 34. D. Verstegen, New exceptional modular invariant partition functions for simple Kac–Moody algebras, Nucl. Phys. B346 (1990), 349–386. 35. M. A. Walton, Algorithm for WZW fusion rules: a proof, Phys. Lett. B241 (1990), 365–368. 36. A. J. Wassermann, Operator algebras and conformal field theory, in: “Proc. ICM, Zurich", Birkhauser, Basel, 1995. 37. E. Witten, The Verlinde formula and the cohomology of the Grassmannian, in: “Geometry, Topology and Physics”, International Press, Cambridge, MA, 1995.	<p>28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products of $$E_{8}$$ representations, Alg., Groups Geom. 3 (1986), 286–328. 29. W. G. McKay, J. Patera and D. W. Rand, “Tables of representations of simple Lie algebras”, Vol. 1, Centre de Recherches Math´ematiques, Univ´ersit´e de Montr´eal, 1990. 30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in: “Interface between physics and mathematics”, World-Scientific, 1994, (W. Nahm and J.-M. Shen, Eds.), World-Scientific, Singapore, 1994. 31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents, Phys. Lett. B244 (1990), 255–260. 32. V. G. Turaev, “Quantum invariants of knots and 3-manifolds”, Walter de Gruyter, Berlin, 1994. 33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. 300 (1988), 360–376. 34. D. Verstegen, New exceptional modular invariant partition functions for simple Kac–Moody algebras, Nucl. Phys. B346 (1990), 349–386. 35. M. A. Walton, Algorithm for WZW fusion rules: a proof, Phys. Lett. B241 (1990), 365–368. 36. A. J. Wassermann, Operator algebras and conformal field theory, in: “Proc. ICM, Zurich", Birkhauser, Basel, 1995. 37. E. Witten, The Verlinde formula and the cohomology of the Grassmannian, in: “Geometry, Topology and Physics”, International Press, Cambridge, MA, 1995.</p>	[{"type": "text", "coordinates": [72, 70, 543, 388], "content": "28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products\nof $$E_{8}$$ representations, Alg., Groups Geom. 3 (1986), 286\u2013328.\n29. W. G. McKay, J. Patera and D. W. Rand, \u201cTables of representations of simple\nLie algebras\u201d, Vol. 1, Centre de Recherches Math\u00b4ematiques, Univ\u00b4ersit\u00b4e de Montr\u00b4eal,\n1990.\n30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in:\n\u201cInterface between physics and mathematics\u201d, World-Scientific, 1994, (W. Nahm and\nJ.-M. Shen, Eds.), World-Scientific, Singapore, 1994.\n31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents,\nPhys. Lett. B244 (1990), 255\u2013260.\n32. V. G. Turaev, \u201cQuantum invariants of knots and 3-manifolds\u201d, Walter de Gruyter,\nBerlin, 1994.\n33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory,\nNucl. Phys. 300 (1988), 360\u2013376.\n34. D. 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