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 $k[Q\,C_G(Q)]$-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.