Character bounds for finite groups of Lie type

TL;DR

This paper derives stronger bounds on character values for finite groups of Lie type, enabling improved estimates for applications like random walk mixing times and covering numbers.

Contribution

It introduces new, sharper bounds on character ratios for finite Lie type groups, many of which are optimal, enhancing understanding of their representation theory.

Findings

New bounds on character values are established.

Bounds lead to improved estimates of mixing times for random walks.

Applications include better understanding of covering numbers and group mixing properties.

Abstract

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $|\chi(g)| \le \chi(1)^{\alpha_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.