A Formula for Plucker Coordinates Associated with a Planar Network

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 non-negative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Plucker 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 Plucker coordinates.