CHARACTER LEVELS AND CHARACTER BOUNDS

MICHAEL LARSEN,ROBERT M. GURALNICK,PHAM HUU TIEP

doi:10.1017/fmp.2019.9

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 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. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnicket al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019,arXiv:1904.08070] by different methods.

Highlights

IntroductionOur method relies crucially on the Deligne–Lusztig theory [C, DM2]
It is well known that the complex irreducible characters of the symmetric group Sn are indexed by partitions λ = (λ1, . . . , λr ) of n
The goal of this paper is to develop the concept of character level for complex irreducible characters of finite, general or special, linear and unitary groups, that is for GLn(q), GUn(q), SLn(q), and SUn(q), where q is a prime power

Summary

Introduction

Our method relies crucially on the Deligne–Lusztig theory [C, DM2] It utilizes several features of finite general and unitary groups G, including the fact that the centralizer CG∗(s) of any semisimple element in the underlying (dual) algebraic group G∗ (of type GLn) is a Levi subgroup, geometric constructions of unipotent representations of GLn(q), and the Ennola duality providing a bridge between the character theories of GLn(q) and GUn(q). These properties allow us to develop a complete character level theory for GLn(q) and GUn(q).

Preliminaries

Character levels in finite general linear and unitary groups

Further results on character levels

Bounds on character degrees

Bounds on character values

Special linear and special unitary groups

Some further results