Arrangements of minors in the positive Grassmannian and a triangulation of the hypersimplex
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Author(s)
Farber, Miriam
Mandelshtam, Yelena
Date Issued
August 2017
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Citation
Farber, Miriam, and Yelena Mandelshtam. “Arrangements of Minors in the Positive Grassmannian and a Triangulation of the Hypersimplex.” Journal of Algebraic Combinatorics 47, no. 3 (August 23, 2017): 473–504.
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Author's final manuscript
Abstract
The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here, we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of a partially ordered set on the entire collection of minors. We use this triangulation of the hypersimplex to describe a two-dimensional grid structure on this poset.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10801-017-0782-2
