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

The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov \cite{Pos} introducing the boundary measurement map in terms of discrete path integration on planar bicoloured (plabic) graphs in the disc. An alternative parametrization was proposed by T. Lam \cite{Lam2} introducing systems of relations at the vertices of such graphs, depending on some signatures defined on their edges. The problem of characterizing the signatures corresponding to the totally non-negative cells, was left open in \cite{Lam2}. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non--negativity property on the full positroid cell. If each edge in a graph $\mathcal G$ belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and the gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of the geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property \cite{AGPR,A3}, and we conjecture a mechanical-statistical interpretation of such relations. An explicit connection between the solution of Lam's system of relations and the value of Postnikov's boundary measurement map is established using the generalization of Talaska's formula \cite{Tal2} obtained in \cite{AG6}. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for the transformations of the signatures under Postnikov's moves and reductions, and amalgamations of networks.