Characters of $GL_n(\mathbb F_q)$ and vertex operators

Abstract

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group $\GL_n(\mathbb F_q)$. Green's theory of $\GL_n(\mathbb F_q)$ is recovered and enhanced under the realization of the Grothendieck ring of representations $R_G=\bigoplus_{n\geq 0}R(\GL_n(\mathbb F_q))$ as two isomorphic Fock spaces associated to two infinite-dimensional $F$-equivariant Heisenberg Lie algebras $\widehat{\mathfrak{h}}_{\hat{\overline{\mathbb F}}_q}$ and $\widehat{\mathfrak{h}}_{\overline{\mathbb F}_q}$, where $F$ is the Frobenius automorphism of the algebraically closed field $\overline{\mathbb F}_q$. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space $R_G$ as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.