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

We study toric degenerations arising from Grobner degen-erations or the tropicalization of partial flag varieties. We produce a new family of toric degenerations of partial flag vari-eties 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 Plucker forms for the Grassmannian. We give a description of matching field polytopes of flag varieties as Minkowski sums and show that all such polytopes are normal. The poly -topes we obtain are examples of Newton-Okounkov bodies for particular full-rank valuations for partial flag varieties. Fur-thermore, 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. (c) 2023 Elsevier Inc. All rights reserved.