Low-degree Cohomology of Integral Specht Modules

TL;DR

This paper investigates the second-degree cohomology of Specht modules over integers and finite fields, providing partial classifications and insights into their prime divisors using algebraic tools.

Contribution

It introduces methods to describe cohomology of symmetric groups with Specht modules over Z and F_p, focusing on second-degree cohomology and partial classifications.

Findings

Exact determination of second-degree cohomology in few cases

Partial information on prime divisors of cohomology groups

Application of Zassenhaus algorithm and branching rules

Abstract

We introduce a way of describing cohomology of the symmetric groups with coefficients in Specht modules over Z or F_p. We study i-th-degree cohomology for i in {0,1,2}. The focus lies on the isomorphism type of second-degree cohomology of integral Specht modules. Unfortunately, only in few cases can we determine this exactly. In many cases we obtain only some information about the prime divisors of the cohomology group's order. The most important tools we use are the Zassenhaus algorithm, the Branching Rules, Bockstein type homomorphisms, and the results from [Burichenko et al., 1996].