VOL. 88 | 2023 Combinatorial mutations of Newton–Okounkov polytopes arising from plabic graphs
Akihiro Higashitani, Yusuke Nakajima

Editor(s) Yukari Ito, Akira Ishii, Osamu Iyama

Adv. Stud. Pure Math., 2023: 227-278 (2023) DOI: 10.2969/aspm/08810227


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.


Published: 1 January 2023
First available in Project Euclid: 8 May 2023

Digital Object Identifier: 10.2969/aspm/08810227

Primary: 14M15
Secondary: 13F60 , 14M25 , 52B20

Keywords: Cluster mutations , Combinatorial mutations , Grassmannians , Newton–Okounkov bodies , Plabic graphs

Rights: Copyright © 2023 Mathematical Society of Japan


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