Recognition of Brauer indecomposability for a Scott module

Abstract

We give a handy way to have a situation that the [Formula: see text]-Scott module with vertex [Formula: see text] remains indecomposable under taking the Brauer construction for any subgroup [Formula: see text] of [Formula: see text] as [Formula: see text]-module, where [Formula: see text] is a field of characteristic [Formula: see text]. The motivation is that the Brauer indecomposability of a [Formula: see text]-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 Scott module fails in Ishioka’s recent examples.