Semitoric degenerations of Hibi varieties and flag varieties

TL;DR

This paper constructs flat semitoric degenerations for Hibi varieties and flag varieties, revealing their component structure and connections to polytopes, with applications to Grassmannians and flag varieties.

Contribution

It introduces a novel family of semitoric degenerations for Hibi and flag varieties, linking their components to polytopal subdivisions and embedding them into toric varieties.

Findings

Components are Hibi varieties associated with polytopal subdivisions.

Constructed degenerations include Grassmannians and flag varieties.

Weight polytopes project onto order polytopes with regular subdivisions.

Abstract

We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we also define the corresponding weight polytope and embed the degeneration into the associated toric variety as the union of orbit closures given by a set of faces. Every such weight polytope projects onto the order polytope with the chosen faces projecting into the parts of the corresponding regular subdivision. We apply these constructions to obtain a family of flat semitoric degenerations for every type A Grassmannian and complete flag variety.