Indecomposable Summands of Foulkes Modules

In this paper we study the modular structure of the permutation module H ) of the symmetric group S2n acting on set partitions of a set of size 2n into n sets each of size 2, defined over a field of odd characteristic p. In particular we characterize the vertices of the indecomposable summands of H ) and fully describe all of its indecomposable summands that lie in blocks of p-weight at most two. When 2n < 3p we show that there is a unique summand of H ) in the principal block of S2n and that this summand exhibits many of the extensions between simple modules in its block.