Combinatorial mutations of Newton–Okounkov polytopes arising from plabic graphs

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.