Structure and geometry of the tableaux algebra

TL;DR

This paper investigates the algebraic structure of the tableaux algebra, revealing its key properties, classifying modules and ideals, and connecting it to geometric objects like flag varieties and crystal embeddings.

Contribution

It provides a comprehensive analysis of the tableaux algebra, including its properties, module classification, and geometric connections, extending previous results on crystal embeddings.

Findings

The algebra is commutative, Noetherian, reduced, Koszul, and Cohen--Macaulay.

Complete classification of maximal ideals and irreducible modules.

Establishment of a connection to toric degenerations of partial flag varieties.

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.