Recognition of Brauer indecomposability for a Scott module

Abstract

We give a handy way to have a situation that the kG-Scott module with vertex P remains indecomposable under taking the Brauer construction for any subgroup Q of P as k[QCG(Q)]-module, where k is a field of characteristic p > 0. The motivation is that the Brauer indecomposability of a p-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.