October 2010, Volume 13, Issue 5, pp 511-519,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 17 Feb 2009
Remarks on the McKay Conjecture
- Geoffrey Mason
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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.
Presented by Jon Carlson.
Supported by grants from the NSF, NSA, and faculty research funds granted by the University of California at Santa Cruz.
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)MathSciNet
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)
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)
Montgomery, S.: Hopf Algebras and Their Actions on Rings, CBMS Number 82. Amer. Math. Soc. (1993)
- Remarks on the McKay Conjecture
- Open Access
- Available under Open Access This content is freely available online to anyone, anywhere at any time.
Algebras and Representation Theory
Volume 13, Issue 5 , pp 511-519
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- McKay correspondence
- Quantum double
- Geoffrey Mason (1)
- Author Affiliations
- 1. Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA, 95064, USA