Vector-relation configurations and plabic graphs

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.

32 pages, 22 figures, to appear in Selecta Mathematica

References in corpus (6)

• Total positivity, Grassmannians, and networks
• Ising model and the positive orthogonal Grassmannian
• Dimers and Circle patterns
• Totally nonnegative Grassmannian and Grassmann polytopes
• Inverse spectral problem for GK integrable system
• Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder

Cited by in corpus (5)

• Dual Graph Convolutional Network for Semantic Segmentation
• Real regular KP divisors on $\mathtt{M}$-curves and totally non-negative Grassmannians
• Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk
• Möbius invariant Y-systems (cluster structures) for Miquel dynamics
• Plabic Tangles and Cluster Promotion Maps