Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs

Abstract

We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr(3, 6).

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Acknowledgments

The main ideas of this paper were developed during the MFO Miniworkshop “PBW-structures in Representation theory”, where all authors enjoyed the hospitality. We would like to thank Peter Littelmann, Diane Maclagan, Markus Reineke, Kristin Shaw, and Bernd Sturmfels for helpful discussions. M.H. would like to thank Daniel Erman, Claudiu Raicu, and Greg Smith for support with Macaulay 2. The work of X.F. was supported by the Alexander von Humboldt foundation. The work of M.H. was partially supported by EPSRC first grant EP/K041002/1. The work of M.L. was funded by the University of Edinburgh.

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Correspondence to M. Lanini.

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Bossinger, L., Fang, X., Fourier, G. et al. Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs. Ann. Comb. 22, 491–512 (2018). https://doi.org/10.1007/s00026-018-0395-z

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Mathematics Subject Classification

  • 14M15
  • 14T05
  • 14M25
  • 05C21

Keywords

  • Grassmannians
  • toric varieties
  • tropical varieties
  • Groebner degenerations
  • plabic graphs