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Conference Papers Year : 2020

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. 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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Dates and versions

hal-03364756 , version 1 (04-10-2021)
hal-03364756 , version 2 (27-10-2022)

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  • HAL Id : hal-03364756 , version 2

Cite

Niklas Affolter, Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy. Vector-relation configurations and plabic graphs (extended abstract). Proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics, Jul 2020, Ramat Gan, Israel. Article #91, 12pp. ⟨hal-03364756v2⟩
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