A formula for Pl\"ucker coordinates associated with a planar network
Combinatorics
2008-09-18 v2 Rings and Algebras
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.
Keywords
Cite
@article{arxiv.0801.4822,
title = {A formula for Pl\"ucker coordinates associated with a planar network},
author = {Kelli Talaska},
journal= {arXiv preprint arXiv:0801.4822},
year = {2008}
}
Comments
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