On permutation modules and decomposition numbers of the symmetric group

Abstract

We study the indecomposable summands of the Foulkes permutation module obtained by inducing the trivial F(Sa≀Sn)-module to the full symmetric group San for any field F of odd prime characteristic p such that a<p≤n. In particular, we characterize the vertices of such indecomposable summands. As a corollary we disprove a modular version of Foulkes' Conjecture.In the second part of the article we use this information to give a new description of some columns of the decomposition matrices of symmetric groups in terms of the ordinary character ϕ(an) of the Foulkes permutation module.