Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs

TL;DR

This paper explores how Newton-Okounkov polytopes associated with Grassmannians, derived from plabic graphs, behave under combinatorial mutations, linking cluster structures, toric degenerations, and polytope transformations.

Contribution

It introduces a framework to understand the combinatorial mutation of Newton-Okounkov polytopes in the context of plabic graph-induced cluster structures.

Findings

Newton-Okounkov polytopes can be mutated combinatorially.

Operations on polytopes correspond to cluster mutations.

The study provides reinterpretations of polytope transformations.

Abstract

It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton--Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article is to survey these phenomena and observe the behavior of Newton--Okounkov polytopes under the operation called the combinatorial mutation of polytopes. In particular, we reinterpret some operations defined for Newton--Okounkov polytopes using the combinatorial mutation.