On the cohomology of Specht modules

AbstractWe investigate the cohomology of the Specht module Sλ for the symmetric group Σd. We show if 0⩽i⩽p−2, then Hi(Σd,Sλ) is isomorphic to Hs+i(B,w0⋅λ′−δ) where s=d(d−1)2, B is the Borel subgroup of the algebraic group GLd(k) and δ=(1d) is the weight of the determinant representation. We obtain similar isomorphisms of ExtΣdi(Sλ,M) with B-cohomology, which in turn yield isomorphisms of cohomology for Borel subgroups of GLn(k) for varying n⩾d. In the case i=0, and the case i=1 for certain λ, we apply our result and known symmetric group results of James and Erdmann to obtain new information about B-cohomology. Finally we show that Specht module cohomology is closely related to cohomology for the Frobenius kernel B1 for small primes