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Séminaire Lotharingien de Combinatoire, 84B.91 (2020), 12 pp.

# Niklas Affolter, Max Glick, Pavlo Pylyavskyy and Sanjay Ramassamy

# 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.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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