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arXiv · 2510.27209

Structure and geometry of the tableaux algebra

Spencer Daugherty·Nicolle González·Bárbara Muniz·Pablo S. Ocal·Jianping Pan·Jacinta Torres
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Abstract

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.

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Spencer Daugherty, Nicolle González, Bárbara Muniz, Pablo S. Ocal, Jianping Pan, Jacinta Torres. 2025-10-31. Structure and geometry of the tableaux algebra. https://arxiv.org/abs/2510.27209

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