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

Recognition of Brauer indecomposability for a Scott module

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 Scot…

提升一个Scott模块的Brauer不可分解性

Lifting Brauer indecomposability of a Scott module

该研究证明了在特定条件下，一个Scott模块的Brauer不可分解性可以提升到更大的群，为相关领域的研究提供了理论支持。

On the Brauer indecomposability of Scott modules

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, …

The Brauer indecomposability of Scott modules and the quadratic group Qd(p)

Let $k$ be an algebraically closed field of prime characteristic $p$ and $P$ a finite $p$-group. We compute the Scott $kG$-module with vertex $P$ when $\mathcal{F}$ is a constrained fusion system on $P$ and $G$ is Park's group for $\mathcal{F}$. In the case $\mathcal{F}$ is a fusion system of the quadratic group $Qd(p)=(\mathbb{Z}/p \times \mathbb{Z}/p)\rtimes {\mathrm{SL}}(2,p)$ on a Sylow $p$-subgroup $P$ of $Qd(p)$ and $G$ is Park's group for $\mathcal{F}$, we prove that the Scott $kG$-module with vertex $P$ is Brauer indecomposable.

The Brauer indecomposability of Scott modules with semidihedral vertex

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 t…

Scott模块的核与布拉uer不可分解性

Kernel of Scott modules and Brauer indecomposability

本文研究了Scott模块的布拉eur不可分解性与其模块核的关系，提出了一般化的判据，并证明在某些情况下，Scott模块在p-局部子群上的不可分解性可以提升到原群上。