Finest positroid subdivisions from maximal weakly separated collections

TL;DR

This paper characterizes when maximal weakly separated collections induce the finest positroid subdivisions of hypersimplices in the positive tropical Grassmannian, revealing their structure and boundary properties.

Contribution

It provides a necessary and sufficient condition for maximal weakly separated collections to form positroid subdivisions, and shows these subdivisions are the finest, answering a key question in the field.

Findings

Characterization of when collections induce positroid subdivisions

All such subdivisions are the finest possible

Boundary preservation of maximal weakly separated collections

Abstract

We adopt a formal and algebraic approach of Early \cite{E2} to study the positive tropical Grassmannian $\operatorname{Trop}^+ Gr_{k,n}$. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in $\rm Trop^+Gr_{k,n}$. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \cite{E1}. Plabic graphs, invented by Postnikov \cite{P}, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian.