A formula for Plücker coordinates associated with a planar network

Details A formula for Plücker coordinates associated with a planar network A formula for Plücker coordinates associated with a planar network

2008-01-31

arxiv.org/abs/0801.4822v2

Int Math Res Notices (2008) Vol. 2008, article ID rnn081, 19 pages, published on July 24, 2008.

Mathematics

Combinatorics

15 pages, 6 figures. Extensive additions, including a generalization for arbitrarily oriented planar graphs and a formula for

Scientific paper

10.1093/imrn/rnn081

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.

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