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19
Higher Bruhat orders and cyclic hyperplane arrangements
 TOPOLOGY
, 1993
"... We study the higher Bruhat orders B(n, k) of Manin & Schechtman [MaS] and characterize them in terms of inversion sets, identify them with the posets ZY(C n+1 ' r,n+l) of uniform extensions of the alternating oriented matroids C n ' r for r: = n—k (that is, with the extensions of a cyclic hyperpl ..."
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Cited by 24 (2 self)
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We study the higher Bruhat orders B(n, k) of Manin & Schechtman [MaS] and characterize them in terms of inversion sets, identify them with the posets ZY(C n+1 ' r,n+l) of uniform extensions of the alternating oriented matroids C n ' r for r: = n—k (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), show that B(n, k) is a lattice for k = 1 and for r < 3, but not in general, show that B(n, k) is ordered by inclusion of inversion sets for k — 1 and for r < 4. However, 2?(8,3) is not ordered by inclusion. This implies that the partial order Bc (n, k) defined by inclusion of inversion sets differs from B(n, k) in general. We show that the proper part of Bc (n, k) is homotopy equivalent to S r ~ 2. Consequently, £(n, k) ~ S r ~ 2 for Jfe = 1 and for r < 4. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for r = 3, and that the proper part is not always homotopy equivalent to S r ( M ' ~ 2.
An Introduction to Hyperplane Arrangements
 Lecture notes, IAS/Park City Mathematics Institute
, 2004
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Lattice Congruences of the Weak Order
 ORDER
, 2004
"... We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence la ..."
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Cited by 18 (8 self)
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We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K ⊆ S, letηK: w ↦ → wK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1 ≡ s for every s ∈ (S − K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of joinirreducibles of the congruence lattice of the weak order.
Lattice congruences, fans and Hopf algebras
 J. Combin. Theory Ser. A
"... Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak or ..."
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Cited by 17 (8 self)
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Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the MalvenutoReutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of patternavoidance. Applying these results, we build the MalvenutoReutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of noncommutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.
Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 16 (4 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
Lattice and Order Properties of the Poset of Regions in a Hyperplane Arrangement
 ALGEBRA UNIVERSALIS
, 2002
"... We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a sup ..."
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Cited by 10 (9 self)
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We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof that the order dimension of the poset of regions (again with respect to a canonical base region) of a supersolvable hyperplane arrangement is equal to the rank of the arrangement. In particular, the order dimension of the weak order on a finite Coxeter group of type A or B is equal to the number of generators. The result for type A (the permutation lattice) was proven previously by Flath [11]. We show that the poset of regions of a simplicial arrangement is a semidistributive lattice, using the previously known result [2] that it is a lattice. A lattice is congruence uniform (or “bounded” in the sense of McKenzie [18]) if and only if it is semidistributive and congruence normal [7]. Caspard, Le Conte de PolyBarbut and Morvan [4] showed that the weak order on a finite Coxeter group is congruence uniform. Inspired by the methods of [4], we characterize congruence normality of a lattice in terms of edgelabelings. This
Generation of oriented matroids – A graph theoretical approach
 Discrete Comput Geom
, 2002
"... We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriente ..."
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Cited by 10 (2 self)
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We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs nally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.
Noncrossing partitions and representations of quivers
 Compos. Math
"... We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yie ..."
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Cited by 9 (0 self)
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We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extensionclosed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition. 1.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t