Geodesic
Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

Voir la notice de l'article provenant de la source Episciences

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 partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset. 
@article{DMTCS_2015_special_285_a13,
     author = {Farber, Miriam and Mandelshtam, Yelena},
     title = {Arrangements {Of} {Minors} {In} {The} {Positive} {Grassmannian} {And} a {Triangulation} of {The} {Hypersimplex}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2469},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2469/}
}
TY  - JOUR
AU  - Farber, Miriam
AU  - Mandelshtam, Yelena
TI  - Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
JO  - Discrete mathematics & theoretical computer science
PY  - 2015
VL  - DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2469/
DO  - 10.46298/dmtcs.2469
LA  - en
ID  - DMTCS_2015_special_285_a13
ER  - 
%0 Journal Article
%A Farber, Miriam
%A Mandelshtam, Yelena
%T Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
%J Discrete mathematics & theoretical computer science
%D 2015
%V DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2469/
%R 10.46298/dmtcs.2469
%G en
%F DMTCS_2015_special_285_a13
Farber, Miriam; Mandelshtam, Yelena. Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2469. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2469/

Cité par Sources :