"Frobenius twists" in the representation theory of the symmetric group

2012/04/04 by David J. Hemmer, Hemmer, David J.
Engineering · Mathematics · #20C30 #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Representation Theory (math.RT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1204.1045

openalex publication_date 2012/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the general linear group GL_n(k) over an algebraically closed field k of characteristic p, there are two types of "twisting" operations that arise naturally on partitions. These are of the form λ→ pλ and λ→ λ+ p^rτ The first comes from the Frobenius twist, and the second arises in various tensor product situations, often from tensoring with the Steinberg module. This paper surveys and adds to an intriguing series of seemingly unrelated symmetric group results where this partition combinatorics arises, but with no structural explanation for it. This includes cohomology of simple, Specht and Young modules, support varieties for Specht modules, homomorphisms between Specht modules, the Mullineux map, p-Kostka numbers and tensor products of Young modules.

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