An extension of Viennot's shadow to rook placements via orbit harmonics
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909891387457536 |
|---|---|
| author | Liu, Jasper M. Zhu, Hai |
| author_facet | Liu, Jasper M. Zhu, Hai |
| contents | For fixed positive integers $n,m,r$, let $\mathrm{Mat}_{n \times m}(\mathbb{C})$ be the affine space of $n \times m$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times m}]$. We define a homogeneous ideal $I_{n,m,r}$, where the graded quotient $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ is obtained from the orbit harmonics deformation of the matrix loci corresponding to all rook placements of size at least $r$. By extending rook placements to elements in $\mathfrak{S}_{n+m-r}$ and applying Viennot's shadow line avatar of the Schensted correspondence, we compute the standard monomial basis of the quotient $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ with respect to diagonal monomial orders. We also determine the graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module structure of $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An extension of Viennot's shadow to rook placements via orbit harmonics Liu, Jasper M. Zhu, Hai Combinatorics Commutative Algebra Representation Theory 05E10, 05E18, 05E40 For fixed positive integers $n,m,r$, let $\mathrm{Mat}_{n \times m}(\mathbb{C})$ be the affine space of $n \times m$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times m}]$. We define a homogeneous ideal $I_{n,m,r}$, where the graded quotient $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ is obtained from the orbit harmonics deformation of the matrix loci corresponding to all rook placements of size at least $r$. By extending rook placements to elements in $\mathfrak{S}_{n+m-r}$ and applying Viennot's shadow line avatar of the Schensted correspondence, we compute the standard monomial basis of the quotient $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ with respect to diagonal monomial orders. We also determine the graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module structure of $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$. |
| title | An extension of Viennot's shadow to rook placements via orbit harmonics |
| topic | Combinatorics Commutative Algebra Representation Theory 05E10, 05E18, 05E40 |
| url | https://arxiv.org/abs/2510.23735 |