Remarks on the McKay Conjecture

Details Remarks on the McKay Conjecture Remarks on the McKay Conjecture

2008-07-22

arxiv.org/abs/0807.3546v1

Mathematics

Representation Theory

Scientific paper

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 \emph{equivalent} to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is \emph{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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