Abstract
The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to p (p a prime) of a finite group G and those of the subgroup N, the normalizer of Sylow p-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is equivalent to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is equivalent to parallel statements about holomorphic orbifolds V G, V N. There is a uniform formulation of MC covering these different situations which involves quantum dimensions of objects in pairs of ribbon fusion categories.
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References
Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three-manifolds, and topological field theory. Comm. Math. Phys. 150(1), 83–107 (1992)
Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
Curtis, C., Reiner, I.: Methods of Representation Theory, vol. 1. Wiley-Interscience, New York (1981)
Drinfeld, V.: Quasi-Hopf algebras. Leningr. Math. J. 1, 1419–1457 (1990)
Dong, C., Mason, G.: Vertex operator algebras and moonshine: a survey. In: Adv. Studies in Pure Math. vol. 24, pp. 101–136. Progress in Algebraic Combinatorics, Math. Soc. of Japan, Tokyo (1996)
Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-quantum groups related to orbifold models. In: Modern Quantum Field Theory (Bombay, 1990), pp. 375–383. World Scientific, Singapore (1991)
Isaacs, M., Malle, G., Navarro, G.: A reduction theorem for the McKay conjecture. Invent. Math. 170, 33–101 (2007)
Kashina, Y., Mason, G., Montgomery, S.: Computing the Frobenius-Schur indicator for Abelian extensions of Hopf algebras. J. Algebra 251, 888–913 (2002)
Mason, G.: The quantum double of a finite group and its role in conformal field theory. In: Groups ’93 Galway/St. Andrews, Lond. Math. Soc. Lect. Notes Ser. 212, vol. 2. CUP (1995)
Mason, G., Ng, S.-H.: Central invariants and Frobenius-Schur indicators for semi-simple quasi-Hopf algebras. Adv. Math. 190, 161–195 (2005)
Montgomery, S.: Hopf Algebras and Their Actions on Rings, CBMS Number 82. Amer. Math. Soc. (1993)
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Supported by grants from the NSF, NSA, and faculty research funds granted by the University of California at Santa Cruz.
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Mason, G. Remarks on the McKay Conjecture. Algebr Represent Theor 13, 511–519 (2010). https://doi.org/10.1007/s10468-009-9132-y
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DOI: https://doi.org/10.1007/s10468-009-9132-y