Chain-order polytopes: toric degenerations, Young tableaux and monomial bases

TL;DR

This paper constructs explicit toric degenerations of flag varieties using marked chain-order polytopes, generalizing known models and introducing pipe dream techniques for broader classes of polytopes and infinite-dimensional cases.

Contribution

It extends toric degeneration constructions to all marked chain-order polytopes of the Gelfand--Tsetlin poset, using pipe dreams and generalizing standard monomial theories.

Findings

Realized toric varieties of MCOPs as Gr"obner degenerations of flag varieties

Generalized pipe dream approach to Pl"ucker coordinate realizations

Introduced semi-infinite pipe dreams for infinite poset polytopes

Abstract

Our first result realizes the toric variety of every marked chain-order polytope (MCOP) of the Gelfand--Tsetlin poset as an explicit Gr\"obner (sagbi) degeneration of the flag variety. This generalizes the Sturmfels/Gonciulea--Lakshmibai/Kogan--Miller construction for the Gelfand--Tsetlin degeneration to the MCOP setting. The key idea of our approach is to use pipe dreams to define realizations of toric varieties in Pl\"ucker coordinates. We then use this approach to generalize two more well-known constructions to arbitrary MCOPs: standard monomial theories such as those given by semistandard Young tableaux and PBW-monomial bases in irreducible representations such as the FFLV bases. In an addendum we introduce the notion of semi-infinite pipe dreams and use it to obtain an infinite family of poset polytopes each providing a toric degeneration of the semi-infinite Grassmannian.