Plethysm and orbit harmonics

Abstract

Let ${\hat{I}}_{(b^a)}$ 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 ${\mathbb{C}}^{\left(\genfrac{}{}{0ex}{}{[n]}{2}\right)}$ with coordinate ring $\mathbb{C}\left[{\mathbf{x}}_{\left(\genfrac{}{}{0ex}{}{[n]}{2}\right)}\right]$. Then, we apply orbit harmonics to ${\hat{I}}_{(2^a)}$ and ${\hat{I}}_{(a^2)}$, 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 $\hat{I}{»}_1$. We further extend this $\hat{I}{»}_1$-separation phenomenon to quotients of ${\mathbb{C}}^{\left(\genfrac{}{}{0ex}{}{[n]}{2}\right)}$ where $n$ is odd. Combining ${\hat{I}}_{(b^a)},{\hat{I}}_{(a^b)}$ 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 ${\hat{I}}_{n,m}$ of unordered set partitions of $[n]$ without blocks of size greater than $m$, yielding a graded ${\mathfrak{S}}_n$-module $R({\hat{I}}_{n,m})$. We determine the standard monomial basis of $R({\hat{I}}_{n,m})$ with respect to any monomial order, as well as its graded character formula.

22 pages