Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

TL;DR

This paper investigates the representation theory of finite simple groups of Lie type, showing that most irreducible representations appear in the conjugation permutation representation and the Steinberg square, with specific exceptions.

Contribution

It establishes that all but one irreducible representation of these groups are contained in the conjugation permutation representation and the Steinberg square, clarifying their structure.

Findings

All irreducible representations are constituents of the conjugation permutation representation, except for a specific case in PSU groups.

Most irreducible representations appear in the tensor square of the Steinberg representation, with a single exception.

The paper precisely characterizes the exceptions for both the permutation and Steinberg representations.

Abstract

Let $G$ be a finite simple group of Lie type, and let $\pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $\pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $St\otimes St$, with the same exceptions as in the previous statement.