Structure and geometry of the tableaux algebra
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910015122571264 |
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| author | Daugherty, Spencer González, Nicolle Muniz, Bárbara Ocal, Pablo S. Pan, Jianping Torres, Jacinta |
| author_facet | Daugherty, Spencer González, Nicolle Muniz, Bárbara Ocal, Pablo S. Pan, Jianping Torres, Jacinta |
| contents | We study the monoid algebra ${}_{n}\mathcal{T}_{m}$ of semistandard Young tableaux, which coincides with the Gelfand--Tsetlin semigroup ring $\mathcal{GT}_{n}$ when $m = n$. Among others, we show that this algebra is commutative, Noetherian, reduced, Koszul, and Cohen--Macaulay. We provide a complete classification of its maximal ideals and compute the topology of its maximal spectrum. Furthermore, we classify its irreducible modules and provide a faithful semisimple representation. We also establish that its associated variety coincides with a toric degeneration of certain partial flag varieties constructed by Gonciulea--Lakshmibai. As an application, we show that this algebra yields injective embeddings of $\mathfrak{sl}_n$-crystals, extending a result of Bossinger--Torres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27209 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure and geometry of the tableaux algebra Daugherty, Spencer González, Nicolle Muniz, Bárbara Ocal, Pablo S. Pan, Jianping Torres, Jacinta Commutative Algebra Combinatorics 13A70, 05E40, 13C05, 05E10, 13C70, 14D06, 05E14, 05E16, 17B37 We study the monoid algebra ${}_{n}\mathcal{T}_{m}$ of semistandard Young tableaux, which coincides with the Gelfand--Tsetlin semigroup ring $\mathcal{GT}_{n}$ when $m = n$. Among others, we show that this algebra is commutative, Noetherian, reduced, Koszul, and Cohen--Macaulay. We provide a complete classification of its maximal ideals and compute the topology of its maximal spectrum. Furthermore, we classify its irreducible modules and provide a faithful semisimple representation. We also establish that its associated variety coincides with a toric degeneration of certain partial flag varieties constructed by Gonciulea--Lakshmibai. As an application, we show that this algebra yields injective embeddings of $\mathfrak{sl}_n$-crystals, extending a result of Bossinger--Torres. |
| title | Structure and geometry of the tableaux algebra |
| topic | Commutative Algebra Combinatorics 13A70, 05E40, 13C05, 05E10, 13C70, 14D06, 05E14, 05E16, 17B37 |
| url | https://arxiv.org/abs/2510.27209 |