# The facets of the matroid polytope and the independent set polytope of a positroid

## TL;DR

This paper investigates the geometric structure of positroids by analyzing their matroid and independent set polytopes, providing new combinatorial criteria for understanding their facets and flats directly from decorated permutations.

## Contribution

It introduces a method to describe bases and independent sets of positroids directly from decorated permutations, bypassing Grassmann necklaces, and provides criteria for flats and concordancy.

## Key findings

- Facets of the matroid polytope are characterized via decorated permutations.
- A criterion for flats of cyclic intervals in positroids is established.
- Applications to checking positroid concordancy are demonstrated.

## Abstract

A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.