Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

TL;DR

This paper characterizes edge signatures on planar bipartite networks representing positroid cells, linking geometric indices to total non-negativity and providing explicit formulas for boundary measurement matrices and transformations.

Contribution

It offers an explicit geometric signature characterization for planar bipartite networks in positroid varieties, generalizing Postnikov's and Talaska's results, with formulas for transformations and internal edge vectors.

Findings

Edge vectors are rational functions with subtraction-free denominators.

Boundary measurement matrices are recovered from edge vectors at boundary sources.

Explicit transformation rules for network modifications are provided.

Abstract

Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and identify total non-negativity via edge signatures. Here we provide an explicit characterization of such signatures on the planar bicolored trivalent directed perfect networks in the disk parametrizing positroid cells $S_M^{TNN}$.To a graph $G$ representing $S_M^{TNN}$, we associate a geometric signature satisfying full rank condition and total non--negativity. Such signature is uniquely identified by geometric indices ruled by orientation and gauge ray direction. The image of this map coincides with that of Postnikov boundary measurement map. We solve the system of geometric relations generalizing Postnikov's and Talaska's results for the boundary edges to the internal edges of the graphs: the edge vector components are rational in the weights with subtraction--free denominators, and have explicit expressions in terms of conservative and edge flows. At boundary sources the edge vectors give the boundary measurement matrix. If $G$ is acyclically orientable, all components are subtraction-free rational in the weights w.r.t. a convenient basis. We provide explicit formulas for the transformation rules w.r.t. changes the orientation, the several gauges of the given network, moves and reductions of networks. We show that the image of the boundary measurement map and the dimer partition functions do not coincide if the graph is not bipartite.