Structure and geometry of the tableaux algebra

We study the monoid algebra ${}_n{\mathcal{T}}_m$ of semistandard Young tableaux, which coincides with the Gelfand–Tsetlin semigroup ring ${\mathcal{G}\mathcal{T}}_n$ 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 ${\mathfrak{s}\mathfrak{l}}_n$-crystals, extending a result of Bossinger–Torres.