The University of Michigan Combinatorics Seminar
For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate cell in the totally nonnegative Grassmannian. We will give an explicit formula for Postnikov's map by expressing each Plücker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. It is then easy to see that this formula generalizes Lindström's classical result for acyclic networks.
If we restrict to a special class of networks corresponding to Le-tableaux, then we can also give an explicit description of the inverse map. This was done for the top dimensional cell by Speyer and Williams. Our formula, which holds for any cell, gives each entry of the Le-tableau as a ratio of certain Plücker variables.