Cohomology and generic cohomology of Specht modules for the symmetric group. (Q734782)

Let \(k\) be an algebraically closed field of characteristic \(p\geq 3\), and consider a symmetric group \(\Sigma_d\) on \(d\) objects. Of interest here is the cohomology of Specht modules over \(k\) (i.e., considered as \(k\Sigma_d\)-modules). For a partition \(\lambda\) of \(d\), let \(S^\lambda\) denote the associated Specht module. While some cohomological results are known for dual Specht modules, little is known for Specht modules beyond degree zero. The approach of this paper is to transfer the problem to an algebraic group setting. Consider the general linear group \(\text{GL}_d(k)\) and the Borel subgroup \(B\) of lower triangular matrices. It is known that cohomology of a Specht module can be identified with extensions over \(\text{GL}_d(k)\) between two induced (or costandard) modules \(H^0(d)\) and \(H^0(\lambda)\) (where by \(d\), we mean the partition \((d,0,\dots,0)\)). The \(\text{GL}_d(k)\)-extensions can be further identified as \(B\)-extensions between \(H^0(d)\) and a one-dimensional \(B\)-module. Using information on the composition factors of \(H^0(d)\) allows the author to recover results of James on degree zero cohomology. More significantly, by using known results on degree one extensions over \(B\), the author is able to obtain degree one cohomology results. First, for a partition \(\lambda\) of \(d\), it is shown that \(H^1(\Sigma_{pd},S^{p\lambda})\simeq H^1(\Sigma_{p^2d},S^{p^2\lambda})\). This is quite interesting since it is a cohomology relationship between different groups. This immediately gives a ``generic cohomology'' result by extending to higher powers of \(p\). A second main result is that if \(p^r>d\), then \(H^1(\Sigma_d,S^{\lambda})\simeq H^1(\Sigma_{d+p^r},S^{\lambda+p^r})\), where \(p^r\) is added to the first part of the partition \(\lambda\). The author concludes with a number of open questions.