Vector-relation configurations and plabic graphs

TL;DR

This paper introduces a geometric model for bipartite graph transformations that unifies various dynamical systems and demonstrates the invertibility of boundary measurements in plabic graphs, revealing underlying cluster algebra structures.

Contribution

It presents a new geometric framework for bipartite graph transformations, connecting dynamical systems with cluster algebras and proving invertibility of boundary measurements for plabic graphs.

Findings

Unified geometric model for bipartite graph transformations

Invertibility of boundary measurement map for plabic graphs

Existence of cluster algebra structures in the studied systems

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.