Covering Irrep(S_n) With Tensor Products and Powers

2020/04/11 by Mark Sellke, Sellke, Mark
Mathematics · #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Random Matrices and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2004.05283

openalex publication_date 2020/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study when a tensor product of irreducible representations of the symmetric group S_n contains all irreducibles as subrepresentations; we say such a tensor product covers Irrep(S_n). Our results show that this behavior is typical. We first give a general sufficient criterion for tensor products to have this property, which holds asymptotically almost surely for constant-sized collections of (Plancherel or uniformly) random irreducibles. We also consider the minimal tensor power of a single fixed irreducible representation needed to cover Irrep(S_n). Here a simple lower bound comes from considering dimensions, and we show it is always tight up to a universal constant factor as was recently conjectured by Liebeck, Shalev, and Tiep.

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