A combinatorial approach to Specht module cohomology

Abstract

For a Specht module S^λfor the symmetric group Σ_d, the cohomology H^i(Σ_d, S^λ) is known only in degree i=0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i=1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally we suggest some general conjectures that might be particularly amenable to proof using this description.