Endotrivial Modules for the General Linear Group in a Nondefining Characteristic

TL;DR

This paper investigates the structure of endotrivial modules for groups between SL(n,q) and GL(n,q), establishing bounds on their torsion-free rank and explicitly determining the group in cases with abelian Sylow p-subgroups, using novel methods.

Contribution

It provides new bounds and explicit descriptions of endotrivial modules for groups related to GL(n,q), introducing a new method for computing kernels in endotrivial module groups.

Findings

Torsion free rank of T(G/Z) is at most one.

Explicit determination of T(G/Z) when Sylow p-subgroup is abelian and nontrivial.

Introduction of a new method by Balmer for kernel computation in endotrivial modules.

Abstract

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.