Vertex representations via finite groups and the McKay correspondence

Abstract

Given a finite group $\Gamma$ and a virtual character $\wt$ on it, we construct a Fock space and associated vertex operators in terms of representation ring of wreath products $\Gamma \sim S_n$. We recover the character tables of wreath products $\Gamma \sim S_n$ by vertex operator calculus. When $\Gamma$ is a finite subgroup of $S{U}_2$, our construction yields a group theoretic realization of the basic representations of the affine and toroidal Lie algebras of $ADE$ type, which can be regarded as a new form of McKay correspondence.

26 pages, one reference added, minor changes, to appear in IMRN

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