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

Abstract

AbstractThe standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] 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 [38] 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 [38]. 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 in the case of bipartite graphs [1, 7], and has a different statistical mechanical interpretation otherwise [6]. 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 [51] obtained in [6]. 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.