Restriction coefficients for partitions with at most three columns

Mitchell Lee

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arXiv · 2506.11837

Restriction coefficients for partitions with at most three columns

Mitchell Lee

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Abstract

Let $r \geq 0$, and let $\lambda$ and $\mu$ be partitions such that $\lambda_1 \leq r + 1$. We present a combinatorial interpretation of the plethysm coefficient $\langle s_\lambda, s_\mu[s_r] \rangle$. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions $\lambda$ with $\lambda_1 \leq 3$, we find a combinatorial interpretation for the multiplicities of the irreducible $\mathfrak{S}_n$-submodules of the Schur module $\mathbb{S}^\lambda \mathbb{C}^n$, considered as an $\mathfrak{S}_n$-module.

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Mitchell Lee. 2025-06-13. Restriction coefficients for partitions with at most three columns. https://arxiv.org/abs/2506.11837

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