The Largest Irreducible Representations of Simple Groups

TL;DR

This paper establishes bounds on the largest irreducible representation degrees of finite simple groups, analyzes their asymptotic behavior, and identifies the Steinberg character as having the largest degree among unipotent characters in groups of Lie type.

Contribution

It provides the first bounds relating the largest and smaller degrees of irreducible representations for finite simple groups and characterizes the Steinberg representation's prominence in Lie type groups.

Findings

Largest degree bounded by smaller degrees

Asymptotic behavior analyzed for Lie type groups

Steinberg character has the largest unipotent degree

Abstract

Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of this largest degree for finite groups of Lie type. Moreover, we show that for groups of Lie type, the Steinberg character has largest degree among all unipotent characters.