Author: David Hemmer

 

Title: Cohomology and generic cohomology of Specht modules for the symmetric group

 

Status: Submitted

 

Abstract: 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} withS^{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.