Restriction coefficients for partitions with at most three columns

Abstract

Let $r\ge 0$, and let $\hat{I}»$ and $\hat{I}¼$ be partitions such that $\hat{I}{»}_1\le r+1$. We present a combinatorial interpretation of the plethysm coefficient $⟨s_{\hat{I}}»,s_{\hat{I}}¼[s_r]⟩$. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions $\hat{I}»$ with $\hat{I}{»}_1\le 3$, we find a combinatorial interpretation for the multiplicities of the irreducible ${\mathfrak{S}}_n$-submodules of the Schur module ${\mathbb{S}}^{\hat{I}}»{\mathbb{C}}^n$, considered as an ${\mathfrak{S}}_n$-module.

14 pages