Relative poset polytopes and semitoric degenerations

TL;DR

This paper introduces relative poset polytopes, a new family of polytopes that describe intermediate degenerations of flag varieties, generalizing previous polytopes and revealing their combinatorial and geometric properties.

Contribution

It provides an in-depth study of relative poset polytopes and their toric varieties, extending the understanding of degenerations of flag varieties and generalizing prior polytope families.

Findings

Relative poset polytopes form a new family of polytopes with key combinatorial properties.

These polytopes describe intermediate degenerations of flag varieties.

The family generalizes previous polytopes studied in the literature.

Abstract

The two best studied toric degenerations of the flag variety are those given by the Gelfand--Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term relative poset polytopes: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity.