Combinatorics of essential sets for positroids
type: Item
instance of: scholarly article
title: Combinatorics of essential sets for positroids
zbMATH Open document ID: 1543.14034
author: Fatemeh Mohammadi
author: Francesca Zaffalon
published in: Journal of Algebra
publication date: 2024-08-15T00:00:00Z
review text: A \textit{positroid} is a matroid \(\mathcal{M}\) which is realizable over \(\mathbb{R}\) and such that there exists a realizing matrix \(M\) for \(\mathcal{M}\) with non-negative maximal minors. Historically, \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}] has identified several families of objects in bijection with positroids, such as bounded affine permutations, plabic graphs, Grassmann necklaces, and Le diagrams. In addition, \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)] introduced essential sets as objects associated to permutations, to explore the degeneracy loci of maps of flagged vector bundles.\N\N In this work, the authors try to introduce a new combinatorial description of positroids via ranked essential sets and illustrate how to obtain information about the positroid's structure from them.\N\NFurthermore, in this paper, the authors extend Fulton's essential sets to bounded affine permutations such that the bijection of the latter with positroids permit them to probe the relationship between them. After that, the authors continue to define connected essential sets and verify that they present a facet description of the positroid polytope, as well as equations defining the positroid variety. Next, the authors focus on a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid.\N\NFinally, the authors conclude this paper by establishing an algorithm to recover the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.
reviewed by: Mehrdad Nasernejad
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 7897488
zbMATH Keywords: positroids
zbMATH Keywords: essential sets
zbMATH Keywords: bounded affine permutations
zbMATH Keywords: matroids
MaRDI profile type: Publication
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DOI: 10.1016/J.JALGEBRA.2024.05.019
description: scientific article; zbMATH DE number 7897488