Recognition of Brauer indecomposability for a Scott module

TL;DR

This paper provides a practical criterion for when the Scott module remains indecomposable after applying the Brauer construction, which is crucial for establishing stable equivalences in modular representation theory.

Contribution

It introduces a new method to determine Brauer indecomposability of Scott modules, clarifying when it holds and explaining previous counterexamples.

Findings

A criterion for Brauer indecomposability of Scott modules.

Insight into why Brauer indecomposability fails in certain cases.

Facilitates the construction of stable equivalences of Morita type.

Abstract

We give a handy way to have a situation that the $kG$-Scott module with vertex $P$ remains indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $p>0$. The motivation is that the Brauer indecomposability of a $p$-permutation bimodule is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method, that then can possibly lift to a splendid derived equivalence. Further our result explains a hidden reason why the Brauer indecomposability of the Scott module fails in Ishioka's recent examples.