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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

Citations:

Zbl 0894.20038
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