# The Brauer indecomposability of Scott modules with semidihedral vertex

## TL;DR

This paper establishes a sufficient condition for the Brauer indecomposability of Scott modules with semidihedral vertices in characteristic 2, extending known results for abelian and dihedral cases, which is crucial for equivalence constructions in modular representation theory.

## Contribution

It generalizes the criteria for Brauer indecomposability of Scott modules to semidihedral vertices, broadening the applicability of these conditions in modular representation theory.

## Key findings

- Provides a sufficient condition for Brauer indecomposability of Scott modules with semidihedral vertices.
- Extends previous results from abelian and dihedral cases to semidihedral groups.
- Facilitates the construction of stable equivalences of Morita type in modular representation theory.

## Abstract

We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under 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 semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact 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 due to Brou\'e, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.