scholarly journals Tropical hyperplane arrangements and oriented matroids

2008 ◽  
Vol 262 (4) ◽  
pp. 795-816 ◽  
Author(s):  
Federico Ardila ◽  
Mike Develin
Keyword(s):  
Hyperplane Arrangements ◽  
Oriented Matroids

scholarly journals Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

2021 ◽  
Author(s):  
Michael Cuntz ◽  
Sophia Elia ◽  
Jean-Philippe Labbé
Keyword(s):  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Weak Order ◽  
Polyhedral Cones ◽  
Normal Lattice ◽  
Poset Of Regions

AbstractA catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

scholarly journals Equivariant Log Concavity and Representation Stability

2021 ◽  
Author(s):  
Jacob P Matherne ◽  
Dane Miyata ◽  
Nicholas Proudfoot ◽  
Eric Ramos
Keyword(s):  
Lie Algebra ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Computer Assisted ◽  
Representation Stability ◽  
Low Degree ◽  
Coxeter Arrangement

Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{s}\mathfrak{l}_n$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree.

scholarly journals Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

2021 ◽  
Vol 274 (1345) ◽  
Author(s):  
Stuart Margolis ◽  
Franco Saliola ◽  
Benjamin Steinberg
Keyword(s):  
Representation Theory ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Regular Band ◽  
Left Regular ◽  
The Face ◽  
Left Regular Band ◽  
Poset Topology ◽  
Left Regular Bands

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

scholarly journals Diameters of Cocircuit Graphs of Oriented Matroids: An Update

10.37236/9653 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ilan Adler ◽  
Jesús A. De Loera ◽  
Steven Klee ◽  
Zhenyang Zhang
Keyword(s):  
Directed Graphs ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Low Rank ◽  
Oriented Matroid ◽  
Computer Search ◽  
Linear Programs ◽  
Large Computer ◽  
Positive Side ◽  
Point Configurations

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key  role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof).  For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false. 

scholarly journals On polytopes and generalizations of the KLT relations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Nikhil Kalyanapuram
Keyword(s):  
Scattering Amplitudes ◽  
Large Class ◽  
Moduli Space ◽  
Differential Forms ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Hyperplane Arrangements ◽  
Number Of Solutions ◽  
Scalar Theories ◽  
Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

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