Simple endotrivial modules for linear, unitary and exceptional groups

TL;DR

This paper classifies simple endotrivial modules for finite special linear, unitary, and exceptional groups, using character theory and lifting results, and applies findings to group rank and block length problems.

Contribution

It provides a classification of simple endotrivial modules for certain groups and applies these results to group rank bounds and block length conjectures.

Findings

Quasi-simple groups with faithful simple endotrivial modules have $\,\ell$-rank at most 2.

Principal blocks of finite simple groups do not have Loewy length 4.

Irreducible characters of special linear and unitary groups exhibit vanishing properties.

Abstract

Motivated by a recent result of Robinson showing that simple endotrivial modules essentially come from quasi-simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of Lie type. Our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods. As one application we prove that the $\ell$-rank of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As a second application we complete the proof that principal blocks of finite simple groups cannot have Loewy length 4, thus answering a question of Koshitani, K\"ulshammer and Sambale. Our results also imply a vanishing result for irreducible characters of special linear and unitary groups.