The Divisibility of $\mathrm{GL}(n, q)$ Character Values

TL;DR

This paper investigates the divisibility properties of character values of the general linear group over finite fields, revealing asymptotic behaviors and bounds depending on the relationship between the field size and divisibility parameter.

Contribution

It provides a detailed analysis of the proportion of irreducible characters with divisible values, including asymptotic results and bounds based on coprimality conditions.

Findings

Proportion tends to 1 as n increases when q and d are coprime.

When q and d are not coprime and g=1, the proportion is at most 1/q.

The study characterizes divisibility patterns of character values in GL(n, q).

Abstract

Let $q$ be a prime power, and $d$ a positive integer. We study the proportion of irreducible characters of $\mathrm{GL}(n,q)$ whose values evaluated on a fixed matrix $g$ are divisible by $d$. As $n$ approaches infinity, this proportion tends to $1$ when $q$ is coprime to $d$. When $q$ and $d$ are not coprime, and $g=1$, this proportion is bounded above by $\frac{1}{q}$.