Plethysm and orbit harmonics

Abstract

Let $\Pi_{(b^a)}$ be the locus of unordered set partitions of $[ab]$ with $a$ blocks of size $b$. We embed unordered set partitions of $[n]$ into the affine space $\mathbb{C}^{\binom{[n]}{2}}$ with coordinate ring $\mathbb{C}\Big[\mathbf{x}_{\binom{[n]}{2}}\Big]$. Then, we apply orbit harmonics to $\Pi_{(2^a)}$ and $\Pi_{(a^2)}$, yielding graded $\mathfrak{S}_{2a}$-modules whose graded character formulae respectively refine the Schur expansions of $h_a[h_2]$ and $h_2[h_a]$ according to $\lambda_1$. We further extend this $\lambda_1$-separation phenomenon to quotients of $\mathbb{C}^{\binom{[n]}{2}}$ where $n$ is odd. Combining $\Pi_{(b^a)},\Pi_{(a^b)}$ and orbit harmonics, we propose a conjecture related to Foulkes' conjecture, and we prove the special case $b=2$. We also apply orbit harmonics to the locus $\Pi_{n,m}$ of unordered set partitions of $[n]$ without blocks of size greater than $m$, yielding a graded $\mathfrak{S}_n$-module $R(\Pi_{n,m})$. We determine the standard monomial basis of $R(\Pi_{n,m})$ with respect to any monomial order, as well as its graded character formula.