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Vector-relation configurations and plabic graphs (extended abstract)

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. Evo- lution for different choices of the graph coincides with many notable dynamical sys- tems 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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https://hal.archives-ouvertes.fr/hal-03364756
Contributor : Sanjay Ramassamy Connect in order to contact the contributor
Submitted on : Monday, October 4, 2021 - 10:15:47 PM
Last modification on : Sunday, June 26, 2022 - 3:14:47 AM
Long-term archiving on: : Wednesday, January 5, 2022 - 7:18:51 PM

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  • HAL Id : hal-03364756, version 1

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Niklas Affolter, Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy. Vector-relation configurations and plabic graphs (extended abstract). 32nd Conference on Formal Power Series and Algebraic Combinatorics, Jul 2020, Ramat Gan, Israel. Article #91, 12pp. ⟨hal-03364756⟩

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