Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections

TL;DR

This paper characterizes positroid cells in the Grassmannian of lines using graphs, enabling computation of their dimensions, boundaries, intersections, and maximal positroids, thus advancing combinatorial understanding of non-negative Grassmannians.

Contribution

It introduces a graph-based characterization of positroids in Gr_{≥0}(2,n) and develops algorithms for computing cell boundaries, intersections, and maximal positroids.

Findings

Graph characterization of positroids in Gr_{≥0}(2,n)

Algorithm for computing cell boundaries and intersections

Method for identifying maximal positroids within a set

Abstract

Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. In particular, positroids enumerate a CW decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov has identified several families of combinatorial objects in bijections with positroids. We will provide yet another characterization of positroids for Gr$_{\geq 0}(2,n)$, the Grassmannians of lines, in terms of certain graphs. We use this characterization to compute the dimension and the boundary of positroid cells. This also leads to a combinatorial description of the intersection of positroid cells, that is easily computable. Our techniques rely on determining different ways to enlarge a given collection of subsets of $\{1,\ldots,n\}$ to represent the dependent sets of a positroid, that is the dependencies among the columns of a matrix with non-negative maximal minors. Furthermore, we provide an algorithm to compute all the maximal positroids contained in a set.