Cohomology and generic cohomology of Specht modules for the symmetric group

2008/03/26 by David J. Hemmer, Hemmer, David J.
Mathematics · #20C30 #20G10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0803.3764

openalex publication_date 2008/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about Hⁱ(Σ_d, S^λ). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht modules equating cohomology of S^pλ with S^p2λ. This is the first theorem we know relating Specht modules S^λand S^pλ. The second result equates cohomology of S^λwith S^{λ+ paμ} for large a.

Related

• A combinatorial approach to Specht module cohomology
• Symmetric group modules with Specht and dual Specht filtrations
• On the cohomology of Young modules for the symmetric group
• Large Dimension Homomorphism Spaces Between Specht Modules for Symmetric Groups
• A combinatorial approach to first degree cohomology of Specht modules