On character values of $GL_n(\mathbb F_q)$

TL;DR

This paper employs vertex operator techniques and Fock space realizations to compute character values of $ ext{GL}_n( extbf{F}_q)$ on unipotent classes, providing a new algebraic approach to understanding their character tables.

Contribution

It introduces a vertex algebraic framework for calculating characters of $ ext{GL}_n( extbf{F}_q)$, including a Murnaghan-Nakayama rule and explicit determination of Steinberg characters.

Findings

Derived a Murnaghan-Nakayama rule for $ ext{GL}_n( extbf{F}_q)$

Realized the Grothendieck ring as Fock spaces

Determined Steinberg characters using vertex algebra methods

Abstract

In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $\phi$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras.