Geometric Nature of Relations on Plabic Graphs and Totally Non-negative Grassmannians

Abstract The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms discrete path integration on planar bicoloured (plabic) graphs disc. An alternative proposed T. Lam [38] systems relations at vertices such graphs, depending some signatures defined their edges. problem characterizing corresponding to cells left open [38]. In our paper we provide an explicit construction signatures, satisfying both full rank condition and total non-negativity property positroid cell. If each edge a graph $\mathcal G$ belongs oriented from boundary, then signature is unique up vertex gauge transformation. Such uniquely identified geometric indices (local winding intersection number) ruled orientation O$ ray direction $\mathfrak l$ G$. Moreover, combinatorial representation showing that every finite face just depends number white it. latter characterization Kasteleyn-type case bipartite [1, 7], has different statistical mechanical interpretation otherwise [6]. connection between solution Lam’s system value Postnikov’s established using generalization Talaska’s formula [51] particular, components vectors are rational weights with subtraction-free denominators. Finally, formulas for transformations under moves reductions amalgamations networks.