The Brauer indecomposability of Scott modules with wreathed $2$-group vertices

TL;DR

This paper establishes a sufficient condition for the Brauer indecomposability of Scott modules with wreathed 2-group vertices, advancing understanding of module stability under subgroup restrictions in modular representation theory.

Contribution

It generalizes previous results by providing conditions for Brauer indecomposability of Scott modules with wreathed 2-group vertices, extending beyond abelian cases.

Findings

Provides a sufficient condition for Brauer indecomposability of Scott modules with wreathed 2-group vertices.

Generalizes known results from abelian to wreathed 2-group vertices.

Facilitates the construction of splendid stable equivalences of Morita type.

Abstract

We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain 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 $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) 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.