On the Brauer Indecomposability of Scott Modules

Abstract

Abstract. 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 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, which then may possibly be lifted to a derived equivalence.

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

“…Therefore by (14), Res N C (M 1 ) = Sc(C, ∆P 0 ), so that by the definition of M 1 , we finally have that Res N C (M(∆Q)) = Sc(C, ∆P 0 ). Since ∆Q acts trivially on M(∆Q), this is the equivalent to say that Res N ∆Q•C (M(∆Q)) = Sc(C, ∆P 0 ).…”

“…Hence, Sc(C, ∆P 0 ) | Res N C (M 1 ) and we can write that (14) Res N C (M 1 ) = Sc(C, ∆P 0 ) X for a kC-module X.…”

“…Thus, (14) and (16), so that X | Ind C ∆P 0 (k). So, let us restrict these kC-modules to P 0 × P 0 .…”

“…Next, let us apply the Brauer construction −(∆P 0 ) to (14). Then (Res N C (M 1 ))(∆P 0 ) = (Sc(C, ∆P 0 ))(∆P 0 ) X(∆P 0 ).…”

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.…”