On the cohomology of Specht modules

We investigate the cohomology of the Specht module S-lambda for the symmetric group Sigma(d). We show if 0 <= i <= p-2, then H-i. (Sigma(d), S-lambda) is isomorphic to Hs+i(B, w(0).lambda(1)-delta) where s = d(d - 1)/2, B is the Borel subgroup of the algebraic group GL(d) (k) and delta = (1d) is the weight of the determinant representation. We obtain similar isomorphisms of Ext(Sigma d)(i) (S-lambda, M) with B-cohomology, which in turn yield isomorphisms of cohomology for Borel subgroups of GL(n) (k) for varying n >= d. In the case i = 0, and the case i = 1 for certain lambda, 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 kemel B-1 for small primes. (c) 2006 Elsevier Inc. All rights reserved.