Noncrossing Duality and the Geometry of Positive Tropical Linear Spaces

TL;DR

This paper develops a new algebraic and polyhedral framework for the positive tropical Grassmannian, revealing a duality that links its fan structure to noncrossing combinatorics and elucidates its metric geometry.

Contribution

It introduces a fundamental tropical duality connecting tropical Plücker vectors and cross-ratios, establishing a global bijection with noncrossing tableaux and analyzing the metric geometry of tropical linear spaces.

Findings

Established a tropical duality linking fan structures to noncrossing fans.

Realized the bounded complex as a subdifferential of a roof function on the hypersimplex.

Identified the planar kinematics weight as governing the complex's diameter.

Abstract

While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural framework for general $k$. Both the global face structure of Trop$_{>0}G(k,n)$ and the internal metric geometry of the tropical linear spaces it parametrizes have remained largely uncharted. This paper develops a systematic algebraic and polyhedral foundation that resolves this gap. The engine of our framework is a fundamental tropical duality, analogous to the duality between cluster variables (or more precisely, their $u$-coordinates) and $\mathbf{g}$-vectors, pairing two families of objects introduced by the first author: the planar basis of tropical Pl\"ucker vectors and the planar cross-ratios on the positive configuration space. We prove that this duality links the fan structure of the positive tropical Grassmannian to the noncrossing fan of Santos, Stump, and Welker, yielding a global bijection between integer points of $Trop_{>0}G(k,n)$ and noncrossing tableaux. We then study how this discrete combinatorial data controls the continuous metric geometry of positive tropical linear spaces. We realize the bounded complex of an integer positive tropical linear space as the subdifferential of a central roof function on the hypersimplex, and use this realization to embed it into a dilate of the fundamental alcoved simplex. The dilation factor, and hence the geometric diameter of the complex, is governed by a single invariant, the planar kinematics ($K) weight, which we show equals the number of columns in the associated noncrossing tableau. The results of this work are applied in our parallel work on scaffolds for higher tropical Grassmannians.