Plethysm and orbit harmonics
Hai Zhu
TL;DR
This work applies the orbit-harmonics framework to loci of unordered set partitions embedded in ${\mathbb{C}}^{\binom{[n]}{2}}$, producing graded ${\mathfrak{S}}_n$-modules whose graded Frobenius characters reveal first-row length separations in plethysms such as $h_a[h_b]$ and $h_b[h_a]$. It provides explicit generators and standard monomial bases for the corresponding quotient rings, derives refined Schur expansions, and extends the constructions to odd $n$ via quotients ${\mathbb{C}}[\mathbf{x}_{\binom{[n]}{2}}]/I(n)$ and $/J(n)$. The paper proves a special case ($b=2$) of a conjecture related to Foulkes’ conjecture and develops a general framework for loci $\Pi_{\lambda}$ and their unions, including a detailed analysis of $R(\Pi_{n,m})$ via forest-based bases with explicit graded-character formulas. Together, these results offer new combinatorial and algebraic models for plethysm, provide tools for understanding the representation theory of symmetric groups in this setting, and suggest avenues for further exploration of orbit-harmonics in plethysm and log-concavity phenomena.
Abstract
Let $Π_{(b^a)}$ 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 $Π_{(2^a)}$ and $Π_{(a^2)}$, 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 $λ_1$. We further extend this $λ_1$-separation phenomenon to quotients of $\mathbb{C}^{\binom{[n]}{2}}$ where $n$ is odd. Combining $Π_{(b^a)},Π_{(a^b)}$ 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 $Π_{n,m}$ of unordered set partitions of $[n]$ without blocks of size greater than $m$, yielding a graded $\mathfrak{S}_n$-module $R(Π_{n,m})$. We determine the standard monomial basis of $R(Π_{n,m})$ with respect to any monomial order, as well as its graded character formula.