Restriction coefficients for partitions with at most three columns

TL;DR

This paper provides a combinatorial interpretation of plethysm coefficients and solves the restriction problem for partitions with up to three columns, enhancing understanding of symmetric group representations.

Contribution

It introduces a combinatorial interpretation for plethysm coefficients and addresses the restriction problem for partitions with at most three columns.

Findings

Combinatorial interpretation of plethysm coefficients

Solution to the restriction problem for partitions with up to three columns

Explicit description of multiplicities in Schur modules

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.