Combinatorial Mutations and Block Diagonal Polytopes

TL;DR

This paper explores how combinatorial mutations of matching field polytopes preserve toric degenerations of Grassmannians and produce new Newton-Okounkov bodies, extending known families of polytopes.

Contribution

It introduces the study of combinatorial mutations of matching field polytopes and shows they preserve toric degenerations, expanding the class of Newton-Okounkov bodies for Grassmannians.

Findings

Mutation preserves toric degenerations of Grassmannians.

Mutated polytopes are Newton-Okounkov bodies for Grassmannians.

Extended the family of block diagonal matching fields.

Abstract

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polytopes arising through mutations are Newton-Okounkov bodies for the Grassmannians with respect to certain full-rank valuations. We produce a large family of such polytopes, extending the family of so-called block diagonal matching fields.