Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Subword complexes in Coxeter groups

Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183-190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.     

Rigidity of Coxeter Groups and Artin Groups

A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given by Bernhard Mühlherr. We show that this example belongs to a general operation of diagram twisting. We show that the Coxeter groups defined by twisted diagrams are isomorphic, and, moreover, that the Artin groups they define are also isomorphic, thus answering a question posed by Charney. Finally, we show a number of Coxeter groups are reflection rigid once twisting is taken into account.     

Coxeter groups,Coxeter monoids and the Bruhat order

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.     

Weak buildings of spherical type

In this article constructions are given for weak buildings of spherical type. The buildings are constructed as point-line geometries according to a pattern found within the appropriate Coxeter complexes. In all the examples constructed here there is at least one point of the geometry on no lines of more than two points. We show that this is true in all weak spherical buildings, and discuss to what further extent all weak buildings follow the given construction.     

Regular polyhedra—old and new

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.This work was supported in part by National Science Foundation Grant MPS74-07547 A01.     

Torsion Free Commutator Subgroups of Generalized Coxeter Groups

In this note we consider generalized Coxeter groups and we study the problem of when their commutator subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the commutator subgroup is torsion free.     

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.     

Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types

The goal of this paper is to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space ${\mathbb{H}^3}$ . Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B?r?czky?CFlorian density upper bound for ??congruent horoball?? packings of ${\mathbb{H}^3}$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.     

Homotopy type of the boolean complex of a Coxeter system

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n−1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.     

Modules of Differential Operators of Order 2 on Coxeter Arrangements

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\) -differential operators as a generalization of the study of the module of \(\mathcal{A}\) -derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”     

Orbifold Stiefel-Whitney Classes of Real Orbifold Vector Bundles over Right-Angled Coxeter Complexes*

The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes(i.e., right-angled Coxeter complexes). Similarly to ordinary Stiefel-Whitney classes, orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties.     

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.     

Coxeter cones and their h-vectors

Coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system. They are shellable subcomplexes of the Coxeter complex, and their h-vectors record the distribution of descents among their chambers. We identify a natural class of “graded” Coxeter cones with the property that their h-vectors are symmetric and unimodal, thereby generalizing recent theorems of Reiner-Welker and Brändén about the Eulerian polynomials of graded partially ordered sets.     

Oriented matroid structures from realized root systems

Journal of Algebraic Combinatorics - This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter system in real vector...     

Bounding reflection length in an affine Coxeter group

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections, and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely, we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on ℝ ⁿ is bounded above by 2n, and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.     

Computations in coxeter groups and related geometric objects

Some algorithms of generating a Coxeter group and the set of cosets of this group by a standard subgroup together with estimates of their complexity are given. Computational problems for the geometries and association schemes connected with Coxeter groups are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7–8, pp. 1055–1060, July–August, 1991.     

Simplex method and groups generated by reflections

A detailed account of the simplex method applied to a class of polyhedra (called Coxeter complexes) is given. The specific geometric properties of these polyhedra enable their use as a testing-ground for comparison of various linear programming algorithms. Applications to sorting problems are given.