Combinatorics of symmetric plabic graphs

TL;DR

This paper explores symmetric plabic graphs, their combinatorial properties, and how they relate to total positivity in the Lagrangian Grassmannian, expanding understanding of symmetric structures in Grassmannian parametrizations.

Contribution

It characterizes symmetric plabic graphs and identifies the subset of the Grassmannian realizable by their symmetric weightings, linking combinatorics and total positivity.

Findings

Characterization of symmetric plabic graphs

Description of Grassmannian subset realizable by symmetric graphs

Connection to total positivity in the Lagrangian Grassmannian

Abstract

A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph $G$ to a positroid cell in some totally nonnegative Grassmannian. In this note, we investigate plabic graphs which are symmetric about a line of reflection, up to reversing the colors of vertices. These symmetric plabic graphs arise naturally in the study of total positivity for the Lagrangian Grassmannian. We characterize various combinatorial objects associated with symmetric plabic graphs, and describe the subset of a Grassmannian which can be realized by symmetric weightings of symmetric plabic graphs.