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

TL;DR

This paper introduces a vertex operator framework to construct and compute all irreducible characters of the general linear group over finite fields, connecting representation theory with symmetric functions and Heisenberg algebras.

Contribution

It provides a novel vertex operator approach that simplifies the construction and computation of characters and character tables for $ ext{GL}_n( extbf{F}_q)$, enhancing Green's theory.

Findings

Realization of irreducible characters via Bernstein vertex operators

Complete character table computation demonstrated

Simplified identification of the Fock space as a Hall algebra

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.