Low-degree cohomology of integral Specht modules.(English)Zbl 1169.20026

Summary: We introduce a way of describing cohomology of the symmetric groups $$\Sigma_n$$ with coefficients in Specht modules. We study $$H^i(\Sigma_n,S_R^\lambda)$$ for $$i\in\{0,1,2\}$$ and $$R=\mathbb{Z},\mathbb{F}_p$$. The focus lies on the isomorphism type of $$H^2(\Sigma_n,S_\mathbb{Z}^\lambda)$$. Unfortunately, only in few cases can we determine this exactly. In many cases we obtain only some information about the prime divisors of $$|H^2(\Sigma_n,S_\mathbb{Z}^\lambda)|$$. The most important tools we use are the Zassenhaus algorithm, the branching rules, Bockstein-type homomorphisms, and the results of V. P. Burichenko, A. S. Kleshchev, and S. Martin [J. Pure Appl. Algebra 112, No. 2, 157-180 (1996; Zbl 0894.20038)].

MSC:

 20J06 Cohomology of groups 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory

Zbl 0894.20038
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