A Formula for Plucker Coordinates Associated with a Planar Network

Abstract

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 Plücker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the nonplanar setting, we show that a similar formula holds for special choices of Plücker coordinates.

Combinatorial formulas for -coordinates in a totally nonnegative Grassmannian

“…Although the complete description of Postnikov's map given in [4] is far more complicated (see [6] for explicit combinatorial formulas), in the special case of Γ -networks, it can be viewed as an instance of the classical formula of Lindström [3]. This formula is usually given in terms of weights of edges; we apply Postnikov's transformation from edge weights to face weights [4] to obtain the following restatement of his definition.…”

The geometry of on-shell diagrams

“…It is straightforward 32 As mentioned in section 4 for planar graphs, this property is also achieved by not adding any sign to the M C matrix. A delicate choice of non-trivial signs is however needed for Plücker coordinates to become sums of contributions from perfect matchings while other nice properties are realized.…”

“…Part of the material presented in this section has previously appeared in the literature, in some cases only for the case of planar graphs [31][32][33]. A key point of this article is that these polytopes are also extremely useful beyond planar graphs.…”

“…All flows associated to a given point in the matroid polytope contribute to the same entries in the boundary measurement matrix. As a result, every perfect matching is mapped to a specific Plücker coordinate [22,[31][32][33]. In summary, each point in the matroid polytope is associated with a single Plücker coordinate, but may get contributions from multiple perfect matchings.…”

On toric degenerations of flag varieties

“…constructed in [85]. Here P G is an index set for a certain set of Plücker coordinates read off from the graph G by a combinatorial rule.…”