On the Brauer indecomposability of Scott modules

TL;DR

This paper establishes conditions under which Scott modules remain indecomposable after Brauer construction, extending previous results to non-abelian p-subgroups, which is important for understanding module equivalences in representation theory.

Contribution

It provides new sufficient conditions for the Brauer indecomposability of Scott modules with non-abelian p-subgroups, generalizing earlier abelian cases.

Findings

Conditions for Brauer indecomposability of Scott modules.

Extension of results from abelian to non-abelian p-subgroups.

Implications for stable and derived equivalences.

Abstract

Let $k$ be an algebraically closed field of prime characteristic $p$, and let $P$ be a $p$-subgroup of a finite group $G$. We give sufficient conditions for the $kG$-Scott module $\mathrm{Sc}(G,P)$ with vertex $P$ to remain indcomposable under the Brauer construction with respect to any subgroup of $P$. This generalizes similar results for the case where $P$ is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a $p$-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, which then may possibly be lifted to a derived equivalence.