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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1908.06959 |
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| _version_ | 1866915096254480384 |
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| author | Affolter, Niklas Glick, Max Pylyavskyy, Pavlo Ramassamy, Sanjay |
| author_facet | Affolter, Niklas Glick, Max Pylyavskyy, Pavlo Ramassamy, Sanjay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_06959 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Vector-relation configurations and plabic graphs Affolter, Niklas Glick, Max Pylyavskyy, Pavlo Ramassamy, Sanjay Combinatorics Dynamical Systems 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. |
| title | Vector-relation configurations and plabic graphs |
| topic | Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/1908.06959 |