Low-Degree Cohomology of Integral Specht Modules

Abstract

We introduce a way of describing cohomology of the symmetric groups $\Sig n$ with coefficients in Specht modules. We study $\HlR i$ for $i \in \{0,1,2\}$ and $R = \Z$, $\Fp$. The focus lies on the isomorphism type of $\Hlz 2$. Unfortunately, only in few cases can we determine this exactly. In many cases we obtain only some information about the prime divisors of $|\Hlz 2|$. The most important tools we use are the Zassenhaus algorithm, the branching rules, Bockstein-type homomorphisms, and the results from Burichenko et al., 1996.