The Brauer Indecomposability of Scott Modules for the Quadratic Group Qd(p)

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 2subgroup 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.

The Brauer indecomposability of Scott modules with semidihedral vertex

“…These theorems in a sense generalize [12,11,13,14], and there are results on Brauer indecomposability of Scott modules also in [15,21].…”

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.…”

“…Recall that each direct summand Ind P 0 ×P 0 c (∆P 0 ) (k) is indecomposable with the vertex c (∆P 0 ) by Green's indecomposability theorem. Hence by the Krulll-Schmidt theorem, there are elements c 1 , • • • , c m ∈ C for some m ≥ 1 such that (18) Res…”

Brauer indecomposability of Scott modules with semidihedral vertex

“…These theorems in a sense generalize [11][12][13][14], and there are results on Brauer indecomposability of Scott modules also in [15,21]. Notation 1.3.…”