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 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.

Real regular KP divisors on $${\texttt {M}}$$-curves and totally non-negative Grassmannians

“…D KP, satisfies the reality and regularity conditions established in [24]. As a consequence, we obtain a direct relation between the total non-negativity property encoded in the geometrical setting of [7,8] and the reality and regularity condition of the divisor studied in this paper.…”

“…In Sect. 5 we perform the first step using the full-rank geometric system of linear relations on the network introduced in [8], after assigning the unnormalized Sato wave function at the marked points κ j as boundary conditions on G. Both the geometric and the weight gauges act on the wave function by multiplication by nonzero constants, therefore they do not affect the normalized wave function. Moreover, it is possible to explicitly solve the system of relations in terms of flows [7,69] and to compute explicitly the KP-II divisor in the coordinates associated to the chosen orientation.…”

“…Such relations were introduced in [52], where the problem of characterizing relations respecting the total non-negativity was posed. In [8] we provided a geometric solution of this problem. Another approach to the same problem was developed in [9] using Kasteleyn's the-orem.…”

“…In [52] it is conjectured that there exist simple rules to assign signatures ε U ,V so that the above system has full rank for any choice of positive weights, and the image of this weighted space of relations is the positroid cell S TNN M ⊂ Gr TNN (k, n) corresponding to the graph. In [8] we provide an explicit construction of such signatures using the geometric indexes introduced in [7], and we call such signatures geometric.…”

“…We remark that our construction of real regular KP-II divisors on M-curves is completely explicit. To this aim we use the full rank geometric system of relations on the network introduced in [7,8] to fix the values of the KP-II wave function at the nodes of the spectral curve. Since such a system of relations provides the value of the Postnikov boundary measurement map for any choice of positive edge weights and is modeled on the amalgamation procedure for totally non-negative Grassmannians [12,26,52], we conjuncture that a purely cluster algebraic approach should be possible for the characterization of the KP-II divisor.…”