Brauer indecomposability of Scott modules with semidihedral vertex

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 by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$ -permutation bimodule (where $p$ 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 Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

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

“…Then the Scott module Sc(G×G ′ , ∆P ) is Brauer indecomposable. This theorem in a sense generalizes [15,14,16,17], and there are results on Brauer indecomposability of Scott modules also in [18,19,26]. Notation 1.2.…”

“…Hence, [25, (27.7) Corollary] implies that Ind P 0 ×P 0 c (∆P 0 ) (k) ( c (∆P 0 )) = 0, so that X( c (∆P 0 )) = 0. Now since X is a kC-module and c ∈ C, we have that 0 = X( c (∆P 0 )) = c X( c (∆P 0 )) = c X(∆P 0 ), so that (19) X(∆P 0 ) = 0.…”

“…By (19), (21) the right hand side of ( 20) is not indecomposable. Now, let us look at the left hand side of (20).…”