Trivial-source endotrivial modules for sporadic groups

TL;DR

This paper classifies endotrivial modules for sporadic simple groups, extending known results, constructing trivial-source modules, and resolving open questions about their existence in many cases.

Contribution

It provides a complete determination of endotrivial modules for sporadic groups and addresses open problems about simple modules being endotrivial.

Findings

Group of endotrivial modules explicitly determined for sporadic groups

Constructed trivial-source endotrivial modules in many cases

Resolved open questions on endotriviality of certain simple modules

Abstract

We determine the group of endotrivial modules (as an abstract group) for $G$ a (quasi)simple group of sporadic type, extending previous results in the literature. In many sporadic cases we directly construct the subgroup of trivial-source endotrivial modules. We also resolve the question of whether certain simple modules for sporadic groups are endotrivial, posed by Lassueur, Malle and Schulte, in the majority of open cases. The results rely heavily on a recent description of the group of trivial-source endotrivial modules due to Grodal.