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

TL;DR

This paper investigates the structure of Scott modules associated with quadratic groups in modular representation theory, demonstrating Brauer indecomposability for specific fusion systems and groups.

Contribution

It computes Scott modules for constrained fusion systems and proves Brauer indecomposability in the case of quadratic groups Qd(p).

Findings

Scott module with vertex P is computed for constrained fusion systems.

Proves Brauer indecomposability of Scott modules for quadratic groups Qd(p).

Results enhance understanding of modular representations of quadratic groups.

Abstract

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.