Positroids and Schubert matroids

TL;DR

This paper proves that positroids are precisely intersections of permuted Schubert matroids, providing a clear combinatorial characterization that enhances understanding and computation within the totally-nonnegative Grassmannian.

Contribution

It confirms Postnikov's conjecture, establishing a new combinatorial framework for positroids as intersections of permuted Schubert matroids.

Findings

Positroids are exactly intersections of permuted Schubert matroids.

Provides a computable combinatorial description of positroids.

Confirms a key conjecture by Postnikov.

Abstract

Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable.