Cohomology and generic cohomology of Specht modules for the symmetric group

David J. Hemmer

Published 2008-03-26, updated 2009-01-28Version 2

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(\Sigma_d, S^\lambda). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht modules equating cohomology of S^{p\lambda} with S^{p^2\lambda}. This is the first theorem we know relating Specht modules S^\lambda and S^{p\lambda}. The second result equates cohomology of S^\lambda with S^{\lambda + p^a\mu} for large a.