Steinberg-like characters for finite simple groups

TL;DR

This paper classifies primes for which finite simple groups admit Steinberg-like characters, extending understanding of special characters related to Sylow subgroups and their representation theory.

Contribution

It determines all primes for which finite simple groups have Steinberg-like characters, except for alternating groups in characteristic 2, and analyzes related projective modules.

Findings

Identified primes with Steinberg-like characters for all finite simple groups except certain alternating groups.

Connected the existence of Steinberg-like characters to the presence of specific projective modules.

Provided a comprehensive classification enhancing the understanding of group characters and modular representations.

Abstract

Let $G$ be a finite group and, for a prime $p$, let $S$ be a Sylow $p$-subgroup of $G$. A character $\chi$ of $G$ is called $\Syl_p$-regular if the restriction of $\chi$ to $S$ is the character of the regular representation of $S$. If, in addition, $\chi$ vanishes at all elements of order divisible by $p$, $\chi$ is said to be Steinberg-like. For every finite simple group $G$ we determine all primes $p$ for which $G$ admits a Steinberg-like character, except for alternating groups in characteristic~2. Moreover, we determine all primes for which $G$ has a projective $FG$-module of dimension $|S|$, where $F$ is an algebraically closed field of characteristic~$p$.