Vector-relation configurations and plabic graphs

• Niklas Affolter ^[1] ; Max Glick ^[4] ; Pavlo Pylyavskyy ^[2] ; Sanjay Ramassamy ^[3]
  1. [1] Technical University of Berlin

     Technical University of Berlin

     Berlin, Stadt, Alemania

     
  2. [2] University of Minnesota

     University of Minnesota

     City of Minneapolis, Estados Unidos

     
  3. [3] University of Paris-Saclay

     University of Paris-Saclay

     Arrondissement de Palaiseau, Francia

     
  4. [4] Google Inc, Pittsburgh, USA

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, 55 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-023-00898-z
• Enlaces
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• Resumen

  • 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. The 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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