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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.
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Contributor : Sanjay Ramassamy Connect in order to contact the contributor
Submitted on : Friday, May 15, 2020 - 8:09:26 PM
Last modification on : Tuesday, January 25, 2022 - 3:09:10 AM


Vectors and relations on bipar...
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  • HAL Id : hal-02595570, version 1


Niklas Affolter, Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy. Vector-relation configurations and plabic graphs. 2020. ⟨hal-02595570⟩



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