Combinatorics of essential sets for positroids
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.
scientific article; zbMATH DE number 7897488
scientific article