Tropicalization of Positive Grassmannians

TL;DR

This paper introduces combinatorial objects called higher laminations to parameterize the positive tropical Grassmannian, extending previous results and linking tropical geometry with configuration spaces of flags and affine buildings.

Contribution

It develops a new combinatorial framework using higher laminations to describe the positive tropical Grassmannian, generalizing earlier specific cases for $Gr(2,n)$ and $Gr(3,6)$, $Gr(3,7).

Findings

Extended tropicalization results to general $Gr(k,n)$.

Connected tropical Grassmannian to configuration spaces of flags.

Provided interpretations of the associated $X$-variety.

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.