Periodicity in the cohomology of finite general linear groups via q-divided powers

TL;DR

This paper demonstrates that the cohomology of finite general linear groups forms a free module over the q-divided power algebra, leading to results on periodicity in cohomology of certain modules and applications to Specht modules.

Contribution

It introduces a novel algebraic structure on the cohomology of GL_n groups and establishes periodicity results for cohomology of ${f VI}$-modules in non-describing characteristic.

Findings

Cohomology admits a q-divided power algebra module structure.

Cohomology is free and generated in degrees ≤ t for q ≠ 2.

Cohomology of ${f VI}$-modules is eventually periodic.

Abstract

We show that $\bigoplus_{n \ge 0} {\mathrm H}^t({\bf GL}_n({\bf F}_q), {\bf F}_\ell)$ canonically admits the structure of a module over the $q$-divided power algebra (assuming $q$ is invertible in ${\bf F}_{\ell}$), and that, as such, it is free and (for $q \neq 2$) generated in degrees $\le t$. As a corollary, we show that the cohomology of a finitely generated ${\bf VI}$-module in non-describing characteristic is eventually periodic in $n$. We apply this to obtain a new result on the cohomology of unipotent Specht modules.