On Plücker coordinates of a perfectly oriented planar network

Kelli Talaska

doi:10.46298/dmtcs.3616

Abstract

Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.

Highlights

Let G be a perfectly oriented planar graph
Postnikov’s groundbreaking paper [7] established combinatorial foundations for the study of totally nonnegative Grassmannians, in particular providing the tools required for the construction of cluster algebra structures in Grassmannians by J
We describe an explicit combinatorial formula for the main construction in [7]: the boundary measurement map assigning a point in the totally nonnegative Grassmannian to a planar directed network with positive edge weights

Summary

Introduction

Let G be a perfectly oriented planar graph. Postnikov’s boundary measurement construction provides a rational map from the set of positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We describe an explicit combinatorial formula for the main construction in [7]: the boundary measurement map assigning a point in the totally nonnegative Grassmannian to a planar directed network with positive edge weights. Postnikov defines the boundary measurement Mij as a weight generating function for directed walks from the boundary vertex bi to the boundary vertex bj, with each walk counted with a sign reflecting the parity of its topological winding index.

Results

Conclusion