Coxeter matroid polytopes

1997 ◽  
Vol 1 (1) ◽  
pp. 123-134 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Israel M. Gelfand ◽  
Neil White

Author(s):  
Luis Ferroni

AbstractWe provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.



1995 ◽  
Vol 13 (3-4) ◽  
pp. 347-361 ◽  
Author(s):  
J. Bokowski ◽  
P. Schuchert
Keyword(s):  


Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.



2020 ◽  
Vol 343 (1) ◽  
pp. 111628
Author(s):  
Suhyung An ◽  
JiYoon Jung ◽  
Sangwook Kim


2008 ◽  
Vol 42 (4) ◽  
pp. 703-704
Author(s):  
Jesús A. De Loera ◽  
David C. Haws ◽  
Matthias Köppe


2010 ◽  
Vol 62 (6) ◽  
pp. 1228-1245 ◽  
Author(s):  
Federico Ardila ◽  
Alex Fink ◽  
Felipe Rincón

AbstractWe prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.



2009 ◽  
Vol 43 (4) ◽  
pp. 841-854 ◽  
Author(s):  
Federico Ardila ◽  
Carolina Benedetti ◽  
Jeffrey Doker
Keyword(s):  


2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Felipe Rincón

International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$.



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