Plethysm and orbit harmonics

TL;DR

This paper explores the application of orbit harmonics to set partition loci, revealing refined symmetric function expansions, proposing a conjecture related to Foulkes' conjecture, and analyzing associated graded modules and bases.

Contribution

It introduces new graded modules from set partition loci using orbit harmonics, refines symmetric function expansions, and proves a special case of a conjecture related to Foulkes' conjecture.

Findings

Refined Schur expansions of symmetric functions via orbit harmonics

Proved a special case of a Foulkes' conjecture-related conjecture

Determined bases and graded characters of modules from set partitions

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.