scholarly journals The face lattice of hyperplane arrangements

1988 ◽  
Vol 73 (1-2) ◽  
pp. 233-238 ◽  
Author(s):  
Günter M. Ziegler
Keyword(s):  
Hyperplane Arrangements ◽  
Face Lattice ◽  
The Face

Lawrence Polytopes

1990 ◽  
Vol 42 (1) ◽  
pp. 62-79 ◽  
Author(s):  
Margaret Bayer ◽  
Bernd Sturmfels
Keyword(s):  
Oriented Matroids ◽  
Oriented Matroid ◽  
Face Lattice ◽  
The Face

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a discussion of this construction and both of these applications.

On the Face Lattice of the Metric Polytope

2003 ◽  
pp. 118-128 ◽  
Author(s):  
Antoine Deza ◽  
Komei Fukuda ◽  
Tomohiko Mizutani ◽  
Cong Vo
Keyword(s):  
Face Lattice ◽  
The Face

scholarly journals Formalizing the Face Lattice of Polyhedra

2020 ◽  
pp. 185-203
Author(s):  
Xavier Allamigeon ◽  
Ricardo D. Katz ◽  
Pierre-Yves Strub
Keyword(s):  
Face Lattice ◽  
The Face

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 Computing the face lattice of a polytope from its vertex-facet incidences

2002 ◽  
Vol 23 (3) ◽  
pp. 281-290 ◽  
Author(s):  
Volker Kaibel ◽  
Marc E. Pfetsch
Keyword(s):  
Face Lattice ◽  
The Face

scholarly journals Affine and toric arrangements

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Richard Ehrenborg ◽  
Margaret Readdy ◽  
Michael Slone
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Face Lattice ◽  
Intersection Lattice ◽  
International Audience

International audience We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

scholarly journals Complementation in the face lattice of a proper cone

1986 ◽  
Vol 79 ◽  
pp. 195-207 ◽  
Author(s):  
Raphael Loewy ◽  
Bit-Shun Tam
Keyword(s):  
Face Lattice ◽  
Proper Cone ◽  
The Face
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