Many zeros of many characters of GL(n,q)

TL;DR

This paper investigates the behavior of the proportion of pairs in the general linear group over finite fields where the character value is non-zero, showing it tends to zero as the dimension grows.

Contribution

It establishes that for the group GL(n,q), the proportion of pairs with non-zero character evaluations diminishes to zero as n increases.

Findings

Proportion P_{n,q} approaches 0 as n increases.

Character evaluations are zero for most pairs in large dimensions.

As n grows, non-zero character pairs become increasingly rare.

Abstract

For $G={\rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi,g)$ in ${\rm Irr}(G)\times G$ with $\chi(g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to\infty$.