Plethysm and orbit harmonics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910021079531520 |
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| author | Zhu, Hai |
| author_facet | Zhu, Hai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12623 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Plethysm and orbit harmonics Zhu, Hai Combinatorics Commutative Algebra Representation Theory 05E10, 05E18, 05E40 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. |
| title | Plethysm and orbit harmonics |
| topic | Combinatorics Commutative Algebra Representation Theory 05E10, 05E18, 05E40 |
| url | https://arxiv.org/abs/2602.12623 |