Stephen Donkin: Invariants of Specht modules

This is joint work with H. Geranios on the the modular representation theory of symmetric groups. The Specht modules for the symmetric group \(S_n\) are characteristic free versions of the ordinary irreducible modules. In particular they are labelled by the partitions of n. Let m and n be positive integers. Then \(S_m\times S_n\) embeds in \(S_{m+n}\) in a natural way so that the space of \(S_m\) invariants of an \(S_{m+n}\)-module is naturally an \(S_n\)-module. We study the module of \(S_m\) invariants of a Specht module \(Sp_L\), where L is a partition of m+n. We show in particular that this module does not always have a filtration by Specht modules, providing a counterexample to a conjecture of D. Hemmer.