Cohomology and generic cohomology of Specht modules for the symmetric group

TL;DR

This paper explores the cohomology of Specht modules for symmetric groups, establishing new stability theorems and linking cohomology of modules with scaled and shifted partitions using algebraic group techniques.

Contribution

It introduces the first known relations between cohomology of Specht modules S^λ and S^{pλ}, and proves generic cohomology stability theorems.

Findings

Recovered classical results for i=0.

Proved stability of cohomology under scaling of partitions.

Established cohomology equivalences for large shifts in partitions.

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} 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.