Vector-relation configurations and plabic graphs

TL;DR

The paper develops a unified geometric framework of vector-relation configurations on bipartite graphs to model local transformations and dynamics, connecting classical systems like the pentagram map, $Q$-nets, and discrete Darboux maps to a common cluster-algebraic structure. It shows that for planar bipartite graphs, and in particular for plabic graphs, boundary data extend uniquely to interior data up to gauge, with the boundary data living in a positroid variety and governed by a reconstruction map that inverts the boundary measurement map on a dense subset. The work integrates Kasteleyn theory, strand diagrams, and cluster dynamics to relate edge-weights, face-weights, and projective data, providing explicit formulas for face weights and a robust geometric interpretation of the boundary measurement map. It also demonstrates the smoothness of the configuration space $\,\\mathcal{C}_G$ and shows how this framework resolves singularities in certain positroid strata, offering new insights into the interplay between geometry, combinatorics, and integrable dynamics with potential applications to resistor networks and the Ising model. Overall, the paper broadens the geometric understanding of discrete integrable systems on networks and clarifies the role of boundary data in reconstructing internal configurations within a cluster-algebraic setting.

Abstract

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. Evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, $Q$-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov's boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for $Q$-nets of whether such a structure exists.

Figures (23)

• Figure 1: A plabic graph corresponding to the open cell in $Gr(3,6)$
• Figure 2: Local transformations applied to a graph.
• Figure 3: The vector-relation version of urban renewal.
• Figure 4: The vector-relation version of degree two vertex addition.
• Figure 5: The evolution equation for face weights

Theorems & Definitions (108)

• Theorem 1.1
• Example 1.2
• Definition 2.1
• Definition 2.2
• Remark 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4