Plabic graphs and dynamical incidence geometry

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. We then introduce a generalized model, where a state consists of a choice of a point for each white vertex and hyperplane for each black vertex. We show that our model behaves consistently under standard local moves of the dimer model. This gives rise to a new class of theorems in linear incidence geometry – dynamical incidence theorems.

Based on joint work with Anton Izosimov, Niklas Affolter, Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy.

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