Zhu's orbit-harmonics strengthening of Foulkes' conjecture: the graded ideal inclusion $\operatorname{gr}\mathbf{I}(\Pi_{(b^a)})\subseteq\operatorname{gr}\mathbf{I}(\Pi_{(a^b)})$

Zhu's Conjecture 3.2 (Hai Zhu, 'Plethysm and orbit harmonics', arXiv:2602.12623, February 2026) states that for positive integers $a\ge b$, the associated graded ideals of the vanishing ideals of the set-partition loci $\Pi_{(b^a)}$ and $\Pi_{(a^b)}$, embedded as $0/1$ points in $\mathbb{C}^{\binom{[ab]}{2}}$, satisfy $\operatorname{gr}\mathbf{I}(\Pi_{(b^a)})\subseteq\operatorname{gr}\mathbf{I}(\Pi_{(a^b)})$. The source paper proves only the case $b=2$. An affirmative answer would induce a graded $\mathfrak{S}_{ab}$-module surjection $R(\Pi_{(b^a)})\twoheadrightarrow R(\Pi_{(a^b)})$ and imply the Schur positivity of $h_a[h_b]-h_b[h_a]$, i.e. Foulkes' conjecture (1950), of which it is a strictly stronger, geometric statement. The conjecture is open for every $b\ge3$.

Background

Let $[n]=\{1,\dots,n\}$. For a partition $\mu$ of $n$, write $\Pi_\mu$ for the set of unordered set partitions of $[n]$ whose block sizes are given by $\mu$; in particular $\Pi_{(b^a)}$ is the family of partitions of $[ab]$ into $a$ blocks of size $b$, a set of size $(ab)!/(b!^a a!)$ carrying an action of the symmetric group $\mathfrak{S}_{ab}$. Zhu (2026) embeds each such set partition as a point of the affine space $\mathbb{C}^{\binom{[n]}{2}}$, whose coordinates $x_S$ are indexed by two-element subsets $S\subseteq[n]$: the coordinate $x_S$ equals $1$ if $S$ lies inside a single block of the partition and $0$ otherwise, so each locus becomes a finite set of $0/1$ vectors.

For a finite locus $Z\subseteq\mathbb{C}^N$, let $\mathbf{I}(Z)$ denote its vanishing ideal, the ideal of polynomials vanishing at every point of $Z$. Orbit harmonics, introduced by Garsia and Procesi (1992) in their study of cohomology rings and $q$-Kostka polynomials, replaces $\mathbf{I}(Z)$ by the associated graded ideal $\operatorname{gr}\mathbf{I}(Z)$: if a nonzero $f\in\mathbf{I}(Z)$ has degree $d$, one keeps its top-degree homogeneous component $f^{(d)}$, and $\operatorname{gr}\mathbf{I}(Z)$ is the ideal generated by all such components. The quotient $R(Z)=\mathbb{C}[\mathbf{x}]/\operatorname{gr}\mathbf{I}(Z)$ is then a graded ring, and there are vector-space (indeed equivariant, when a group stabilizes $Z$) isomorphisms $\mathbb{C}[Z]\cong\mathbb{C}[\mathbf{x}]/\mathbf{I}(Z)\cong R(Z)$; thus $R(Z)$ is a graded refinement of the permutation representation on $Z$.

The representation-theoretic dictionary is expressed through the Frobenius characteristic, which sends a finite $\mathfrak{S}_n$-module to a symmetric function of degree $n$ and maps irreducible Specht modules to Schur functions $s_\lambda$; a symmetric function is Schur positive if all coefficients in its expansion in the $s_\lambda$ basis are nonnegative. Plethysm $f[g]$ is the composition operation on symmetric functions induced by substituting the monomials of $g$ into the variables of $f$. The Frobenius characteristic of the permutation module on $\Pi_{(b^a)}$ (equivalently, of the trivial representation of the wreath product $\mathfrak{S}_b\wr\mathfrak{S}_a$ induced to $\mathfrak{S}_{ab}$) is the plethysm $h_a[h_b]$, where $h_m$ is the complete homogeneous symmetric function. Foulkes' conjecture, posed in 1950 within an invariant-theoretic study of concomitants of the quintic and sextic, asserts that for $a\le b$ the difference $h_b[h_a]-h_a[h_b]$ is Schur positive. It is one of the oldest open problems on plethysm. The known partial results are: the case $a=2$, essentially contained in Thrall's 1942 decomposition $h_2[h_b]=\sum_d s_{(2b-2d,2d)}$; the case $a=3$ (Dent–Siemons, 2000); the case $a=5$, by computer calculation plus a propagation theorem (Cheung–Ikenmeyer–Mkrtchyan, 2017), extending earlier work that had settled $a\le4$; Brion's stable theorem (1993), which proves Schur positivity of $h_b[h_a]-h_a[h_b]$ for fixed $a$ and $b$ sufficiently large; and the strengthened Foulkes conjecture for stable plethysm coefficients, proved by Bowman–Paget (2024) via the partition algebra. The general finite statement remains open.

Zhu (2026) computes $\operatorname{Frob}(R(\Pi_{(b^a)}))=h_a[h_b]$, so the orbit-harmonics refinement carries the same representation theory as the Foulkes module but with a grading. For $b=2$ the picture is complete: $\operatorname{gr}\mathbf{I}(\Pi_{(2^a)})$ has an explicit quadratic generating set (Theorem 3.3, revisiting the $\lambda_1$-refinement phenomenon for involution and matrix loci of Liu–Ma–Rhoades–Zhu and Rhoades), the inclusion $\operatorname{gr}\mathbf{I}(\Pi_{(2^a)})\subseteq\operatorname{gr}\mathbf{I}(\Pi_{(a^2)})$ holds (Theorem 3.4), and consequently $\operatorname{grFrob}(R(\Pi_{(a^2)});q)=\sum_{d=0}^{\lfloor a/2\rfloor}q^d s_{(2a-2d,2d)}$ (Theorem 3.5), a graded refinement of Thrall's formula. Zhu then proposes, as a strictly stronger, algebraic-geometric form of Foulkes' conjecture, the inclusion $\operatorname{gr}\mathbf{I}(\Pi_{(b^a)})\subseteq\operatorname{gr}\mathbf{I}(\Pi_{(a^b)})$ for all $a\ge b$ (Conjecture 3.2), of which only $b=2$ is proved. Notably, even the ingredients needed to attack the conjecture — an explicit generating set and graded character formula for $R(\Pi_{(b^a)})$ when $a,b\ge3$ — are posed as the paper's own open Problem 4.1.

Problem Statement

Prove or disprove Zhu's Conjecture 3.2: for all positive integers $a\ge b$, the associated graded ideals of the vanishing ideals of the set-partition loci embedded in $\mathbb{C}^{\binom{[ab]}{2}}$ satisfy $\operatorname{gr}\mathbf{I}\bigl(\Pi_{(b^a)}\bigr)\subseteq\operatorname{gr}\mathbf{I}\bigl(\Pi_{(a^b)}\bigr)$ as ideals of the polynomial ring $\mathbb{C}\bigl[\mathbf{x}_{\binom{[ab]}{2}}\bigr]$ on the two-element subsets of $[ab]$. Such an inclusion would induce a canonical surjection of graded $\mathfrak{S}_{ab}$-modules $R\bigl(\Pi_{(b^a)}\bigr)\twoheadrightarrow R\bigl(\Pi_{(a^b)}\bigr)$. The case $b=2$ is known (Zhu, Theorem 3.4, building on the explicit generating set of $\operatorname{gr}\mathbf{I}(\Pi_{(2^a)})$ from Theorem 3.3); the conjecture is open for every $b\ge3$. An affirmative answer in full generality would yield $h_a[h_b]\ge h_b[h_a]$ coefficient-wise for all $a\ge b$, that is, Foulkes' conjecture; a single explicit pair $(a,b)$ with $a\ge b\ge3$ and a polynomial in $\operatorname{gr}\mathbf{I}(\Pi_{(b^a)})\setminus\operatorname{gr}\mathbf{I}(\Pi_{(a^b)})$ would refute it.

References

1. Hai Zhu (2026). Plethysm and orbit harmonics. arXiv:2602.12623, doi:10.48550/arXiv.2602.12623, https://arxiv.org/abs/2602.12623
2. H. O. Foulkes (1950). Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society s1-25(3), 205-209, doi:10.1112/jlms/s1-25.3.205, https://doi.org/10.1112/jlms/s1-25.3.205
3. R. M. Thrall (1942). On symmetrized Kronecker powers and the structure of the free Lie ring. American Journal of Mathematics 64(2), 371-388, doi:10.2307/2371691, https://doi.org/10.2307/2371691
4. A. M. Garsia and C. Procesi (1992). On certain graded S_n-modules and the q-Kostka polynomials. Advances in Mathematics 94(1), 82-138, doi:10.1016/0001-8708(92)90034-I, https://doi.org/10.1016/0001-8708(92)90034-I
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