Journal of Combinatorics

Volume 13 (2022)

Number 4

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

Pages: 545 – 560

DOI: https://dx.doi.org/10.4310/JOC.2022.v13.n4.a5

Authors

Suho Oh (Texas State University, San Marcos, Tx., U.S.A.)

David Xiang (Harvard University, Cambridge, Massachusetts, U.S.A.)

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 [14]. 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.

Keywords

positroid, matroid polytope, flacet, non-crossing partition, decorated permutation

2010 Mathematics Subject Classification

Primary 05A05, 52B40. Secondary 05A18, 52B05.

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Received 16 May 2019

Accepted 17 August 2021

Published 18 August 2022