A formula for Pl\"ucker coordinates associated with a planar network

TL;DR

This paper provides an explicit combinatorial formula for Plücker coordinates associated with planar networks, enhancing understanding of the boundary measurement map in the totally nonnegative Grassmannian.

Contribution

It introduces a new explicit formula expressing Plücker coordinates as ratios of positive integer coefficient polynomials in edge weights for planar graphs.

Findings

Explicit formula for Plücker coordinates in planar networks

Extension of the formula to certain non-planar cases

Enhanced combinatorial understanding of the boundary measurement map

Abstract

For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Pluecker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the non-planar setting, we show that a similar formula holds for special choices of Pluecker coordinates.