Tropicalization of positive Grassmannians

Ian Le*, Chris Fraser

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian Gr (k, n). Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov and Shen, configuration spaces of flags naturally tropicalize to give configurations of points in the affine building, which we call higher laminations. We use higher laminations to give two dual objects that are parameterized by the positive tropicalization of Gr (k, n) : equivalence classes of higher laminations; or certain restricted subset of higher laminations. This extends results of Speyer and Sturmfels on the tropicalization of Gr (2 , n) , and of Speyer and Williams on the tropicalization of Gr (3 , 6) and Gr (3 , 7). We also analyze the X-variety associated to the Grassmannian, and give an interpretation of its positive tropicalization.

Original languageEnglish (US)
Article number75
JournalSelecta Mathematica, New Series
Volume25
Issue number5
DOIs
StatePublished - Dec 1 2019
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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