Vertex representations via finite groups and the McKay correspondence

Abstract

Given a finite group $Γ$ and a virtual character $\wt$ on it, we construct a Fock space and associated vertex operators in terms of representation ring of wreath products $Γ\sim S_n$. We recover the character tables of wreath products $Γ\sim S_n$ by vertex operator calculus. When $Γ$ is a finite subgroup of $SU_2$, 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.