Remarks on the McKay Conjecture

TL;DR

This paper explores the implications and equivalences of the McKay Conjecture across various algebraic structures, including crossed products, quantum doubles, and orbifold conformal field theories, unifying these perspectives.

Contribution

It demonstrates the equivalence of the McKay Conjecture with analogous statements in different algebraic and physical contexts, providing a unified formulation involving quantum dimensions.

Findings

MC implies bijections between representations of related algebras.

MC is equivalent to statements about quantum doubles and orbifold theories.

A uniform formulation involves quantum dimensions in ribbon fusion categories.

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 \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.