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 $\mathcal{GT}_{n}$ 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.