Indecomposable summands of Foulkes modules

In this paper we study the modular structure of the permutation module H(2n)H(2n) of the symmetric group S2nS2n 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 characterise the vertices of the indecomposable summands of H(2n)H(2n) and fully describe all of its indecomposable summands that lie in blocks of p  -weight at most two. When 2n<3p2n<3p we show that there is a unique summand of H(2n)H(2n) in the principal block of S2nS2n and that this summand exhibits many of the extensions between simple modules in its block.