Restriction coefficients for partitions with at most three columns

2025/06/13 by Lee, Mitchell
#05E05 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.11837

Abstract

Let r ≥ 0, and let λ and μ be partitions such that λ₁ ≤ r + 1. We present a combinatorial interpretation of the plethysm coefficient ⟨ s_λ, s_μ[s_r] ⟩. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions λ with λ₁ ≤ 3, we find a combinatorial interpretation for the multiplicities of the irreducible \mathfrakS_n-submodules of the Schur module \mathbbS^λℂⁿ, considered as an \mathfrakS_n-module.

Citations

• The Frobenius transform of a symmetric function
• Some Restriction Coefficients for the Trivial and Sign Representations
• A combinatorial model for the decomposition of multivariate polynomial\n rings as S_n-modules
• The Hopf structure of symmetric group characters as symmetric functions
• Hopf Algebras in Combinatorics
• A computational and combinatorial exposé of plethystic calculus

Related

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• 3-Plethysms of homogeneous and elementary symmetric functions
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• Hook restriction coefficients