Two stability theorems on plethysms of Schur functions

2025/05/09 by Rowena Paget, Mark Wildon, Paget, Rowena +1
Computer Science · Mathematics · #20G05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Polynomial and algebraic computation #Primary: 05E05 #Representation Theory (math.RT) #Secondary: 05E10

paper · doi:10.48550/arxiv.2505.06104

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients ⟨ s_ν∘ s_μ, s_λ⟩ that express an arbitrary plethysm s_ν∘ s_μ as a sum ∑_λ⟨ s_ν∘ s_μ, s_λ⟩ s_λ of Schur functions is a fundamental open problem in algebraic combinatorics. We prove two stability theorems for plethysm coefficients under the operations of adding and/or joining an arbitrary partition to either μ or ν. In both theorems μ may be replaced with an arbitrary skew partition. As special cases we obtain all stability results on the plethysm product of two Schur functions in the literature to date. The proofs are entirely combinatorial using plethystic semistandard tableaux with positive and negative entries.

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