Newton-Okounkov bodies obtained from certain orbits of plabic graphs

TL;DR

This paper explores the connection between plabic graphs, dihedral group actions, and Newton-Okounkov bodies, revealing their unimodular equivalence to Gelfand-Tsetlin polytopes in the context of Grassmannian cluster structures.

Contribution

It characterizes the orbits of certain plabic graphs under dihedral symmetry and shows their Newton-Okounkov bodies are unimodularly equivalent to Gelfand-Tsetlin polytopes.

Findings

Newton-Okounkov bodies are unimodularly equivalent to Gelfand-Tsetlin polytopes.

Orbits of plabic graphs under dihedral group actions are explicitly described.

Cluster structures on Grassmannians are linked to these geometric objects.

Abstract

We investigate the plabic graphs corresponding to the quadrilateral Postnikov arrangements used by J.Scott to equip the homogeneous coordinate rings of Grassmannians with a cluster structure. More precisely we describe their orbits under the natural action of the dihedral group and show that the associated Newton-Okounkov bodies are all unimodular equivalent to Gelfand-Tsetlin polytopes.