Character Levels and Character Bounds

TL;DR

This paper introduces the concept of character level for finite linear and unitary groups, providing characterizations, bounds on character values, and applications to random walk mixing times.

Contribution

It defines character levels for complex irreducible characters and derives explicit bounds and characterizations related to these levels, advancing understanding of character behavior.

Findings

Character levels are characterized via Lusztig's labels and degrees.

Explicit upper bounds for character values at certain elements are established.

Bounds on covering numbers and mixing times for related random walks are derived.

Abstract

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large.