Plabic R-matrices

TL;DR

This paper introduces plabic R-matrices, semi-local transformations of weights on cylindrical plabic networks that preserve boundary measurements and reveal underlying cluster algebra structures, extending previous theories.

Contribution

It provides a comprehensive study of plabic R-matrices on cylinders, connecting them to cluster algebras and generalizing recent related work.

Findings

Plabic R-matrices preserve boundary measurements on cylindrical networks.

They have an underlying cluster algebra structure.

Special cases include geometric R-matrices and transformations from Goncharov-Shen.

Abstract

Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this paper we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing recent work of Inoue-Lam-Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein-Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov-Shen.