Tropicalization of positive Grassmannians

Abstract

We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $$\mathrm{Gr}(k,n)$$. 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 $$\mathrm{Gr}(k,n)$$: equivalence classes of higher laminations; or certain restricted subset of higher laminations. This extends results of Speyer and Sturmfels on the tropicalization of $$\mathrm{Gr}(2,n)$$, and of Speyer and Williams on the tropicalization of $$\mathrm{Gr}(3,6)$$ and $$\mathrm{Gr}(3,7)$$. We also analyze the $${\mathcal {X}}$$-variety associated to the Grassmannian, and give an interpretation of its positive tropicalization.