Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes

TL;DR

This paper introduces new toric degenerations of partial flag varieties using matching fields and combinatorial mutations, providing explicit polytope descriptions and demonstrating their normality and connection via mutations.

Contribution

It develops a novel family of toric degenerations for partial flag varieties, characterized by matching field polytopes and their mutation relations, with explicit polytope descriptions.

Findings

All constructed polytopes are normal.

Matching field polytopes are described as Minkowski sums.

Explicit toric degenerations of small Grassmannians and flag varieties are computed.

Abstract

We study toric degenerations arising from Gr\"obner degenerations or the tropicalization of partial flag varieties. We produce a new family of toric degenerations of partial flag varieties whose combinatorics are governed by matching fields and combinatorial mutations of polytopes. We provide an explicit description of the polytopes associated with the resulting toric varieties in terms of matching field polytopes. These polytopes encode the combinatorial data of monomial degenerations of Pl\"ucker forms for the Grassmannians. We give a description of matching field polytopes of flag varieties as Minkowski sums and show that all such polytopes are normal. The polytopes we obtain are examples of Newton-Okounkov bodies for particular full-rank valuations for flag varieties. Furthermore, we study a certain explicitly-defined large family of matching field polytopes and prove that all polytopes in this family are connected by combinatorial mutations. Finally, we apply our methods to explicitly compute toric degenerations of small Grassmannians and flag varieties and obtain new families of toric degenerations.