A formula for Plücker 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 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.

15 pages, 6 figures. Extensive additions, including a generalization for arbitrarily oriented planar graphs and a formula for some Pluecker coordinates of non-planar perfectly oriented graphs