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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.07850 |
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| _version_ | 1866915082425860096 |
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| author | Liu, Jasper M. |
| author_facet | Liu, Jasper M. |
| contents | Let $\mathbf{x}_{n \times n}$ be a matrix of $n \times n$ variables, and let $\mathbb{C}[\mathbf{x}_{n \times n}]$ be the polynomial ring on these variables. Let $\mathfrak{S}_{n,r}$ be the group of colored permutations, consisting of $n \times n$ complex matrices with exactly one nonzero entry in each row and column, where each nonzero entry is an $r$-th root of unity. We associate an ideal $I_{\mathfrak{S}_{n,r}} \subseteq \mathbb{C}[\mathbf{x}_{n \times n}]$ with the group $\mathfrak{S}_{n,r}$, and use orbit harmonics to give an ideal-theoretic extension of the Viennot shadow line construction to $\mathfrak{S}_{n,r}$. This extension gives a standard monomial basis of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$, and introduces an analogous definition of ``longest increasing subsequence'' to the group $\mathfrak{S}_{n,r}$. We examine the extension of Chen's conjecture to this analogy. We also study the structure of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$ as a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module, which subsequently induces a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module structure on the $\mathbb{C}$-algebra $\mathbb{C}[\mathfrak{S}_{n,r}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07850 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Viennot shadows and graded module structure in colored permutation groups Liu, Jasper M. Combinatorics Let $\mathbf{x}_{n \times n}$ be a matrix of $n \times n$ variables, and let $\mathbb{C}[\mathbf{x}_{n \times n}]$ be the polynomial ring on these variables. Let $\mathfrak{S}_{n,r}$ be the group of colored permutations, consisting of $n \times n$ complex matrices with exactly one nonzero entry in each row and column, where each nonzero entry is an $r$-th root of unity. We associate an ideal $I_{\mathfrak{S}_{n,r}} \subseteq \mathbb{C}[\mathbf{x}_{n \times n}]$ with the group $\mathfrak{S}_{n,r}$, and use orbit harmonics to give an ideal-theoretic extension of the Viennot shadow line construction to $\mathfrak{S}_{n,r}$. This extension gives a standard monomial basis of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$, and introduces an analogous definition of ``longest increasing subsequence'' to the group $\mathfrak{S}_{n,r}$. We examine the extension of Chen's conjecture to this analogy. We also study the structure of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$ as a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module, which subsequently induces a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module structure on the $\mathbb{C}$-algebra $\mathbb{C}[\mathfrak{S}_{n,r}]$. |
| title | Viennot shadows and graded module structure in colored permutation groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.07850 |