A Formula for Plücker Coordinates Associated with a Planar Network

By Kelli Talaska. 2008

66 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1f14dcbea0540bed387b3c3dcbee8a59e776f054

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Cluster varieties from Legendrian knots (2015, methodology): “otivation for considering reduced plabic graphs in [Pos] is that each gives rise to a boundary measurement map B : Loc fr 1 () ,! r: We recall the deﬁnition of B as reformulated by the main result of [Tal]. Fix a cyclically-ordered labeling of the boundary vertices of by f1;:::;ng. Orient so that ev”
The twist for positroid varieties (2016, background): “In the language of loop erased walks, Postnikov [27, Theorem 4.11] says that, for every perfectly oriented planar graph H, there is a positroid whose nonzero coordinates are targets of loop erased walks in H and [27, Theorem 4.12] says that every positroid is the nonzero targets of the loop erased w”