Properties of the Coxeter transformations for the affine Dynkin cycle

We study spectral properties of the Coxeter transformations for the affine Dynkin cycle and find the Jordan form of the Coxeter transformation and the Coxeter numbers.     

Combinatorial properties of the Temperley–Lieb algebra of a Coxeter group

We study two families of polynomials that play the same role in the Temperley–Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R-polynomials play in the Hecke algebra of the group. Our results include recursions, non-recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non-branching Coxeter graphs.     

An Adjacency Criterion for Coxeter Matroids

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A _n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.     

On parabolic subgroups in Coxeter groups of large type

In this paper, properties of the elements of parabolic subgroups in Coxeter groups of large type are considered.     

Alternating subgroups of Coxeter groups

We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.     

Walls in Coxeter complexes

The starting point of this paper was the following question: Which walls in Coxeter complexes are Coxeter complexes in their own right? A complete answer to this question is given in the case of finite Coxeter complexes. In general, a sufficient criterion (depending on the entries of the Coxeter matrix) is derived which implies that walls of a certain type are always Coxeter complexes. It is studied how their Weyl groups are related to those of the original Coxeter complexes. Additionally, some statements being true for all walls are proved more generally for convex subcomplexes of Coxeter complexes.     

Continuous crystal and Duistermaat-Heckman measure for Coxeter groups

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations.     

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.     

Coxeter groups are virtually special

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.     

Geometric combinatorics of Weyl groupoids

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.     

Constructions arborescentes d'immeubles

 We prove that, contrarily to the case of spherical and euclidean buidings, the set of (isomorphism classes of) locally finite 3-dimensional hyperbolic buildings is uncountable. The proof uses on one hand a classification of 3-dimensional Coxeter polytops satisfying some local properties of irreducibility and symmetry, and on another hand, an arborescent construction of buildings for splitable Coxeter systems. Received: 20 September 2001 / Revised version: 22 May 2002 / Published online: 2 December 2002 Mathematics Subject Classification (2000): 51E24, 51M10, 51F15     

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.     

The size of a hyperbolic Coxeter simplex

We determine the covolumes of all hyperbolic Coxeter simplex reflection groups. These groups exist up to dimension 9. the volume computations involve several different methods according to the parity of dimension, subgroup relations and arithmeticity properties.     

Coxeter complexes and graph-associahedra

Given a graph Γ, we construct a simple, convex polytope, dubbed graph-associahedra, whose face poset is based on the connected subgraphs of Γ. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford compactification of the real moduli space of curves.     

The Enumeration of Coxeter Elements

Let (W,S, ) be a Coxeter system: a Coxeter group W with S its distinguished generator set and its Coxeter graph. In the present paper, we always assume that the cardinality l=|S| ofS is finite. A Coxeter element of W is by definition a product of all generators s S in any fixed order. We use the notation C(W) to denote the set of all the Coxeter elements in W. These elements play an important role in the theory of Coxeter groups, e.g., the determination of polynomial invariants, the Poincaré polynomial, the Coxeter number and the group order of W (see [1–5] for example). They are also important in representation theory (see [6]). In the present paper, we show that the set C(W) is in one-to-one correspondence with the setC() of all acyclic orientations of . Then we use some graph-theoretic tricks to compute the cardinality c(W) of the setC(W) for any Coxeter group W. We deduce a recurrence formula for this number. Furthermore, we obtain some direct formulae of c(W) for a large family of Coxeter groups, which include all the finite, affine and hyperbolic Coxeter groups.The content of the paper is organized as below. In Section 1, we discuss some properties of Coxeter elements for simplifying the computation of the value c(W). In particular, we establish a bijection between the sets C(W) andC() . Then among the other results, we give a recurrence formula of c(W) in Section 2. Subsequently we deduce some closed formulae of c(W) for certain families of Coxeter groups in Section 3.     

Reflection Independence in Even Coxeter Groups

If (W,S) is a Coxeter system, then an element of W is a reflection if it is conjugate to some element of S. To each Coxeter system there is an associated Coxeter diagram. A Coxeter system is called reflection preserving if every automorphism of W preserves reflections in this Coxeter system. As a direct application of our main theorem, we classify all reflection preserving even Coxeter systems. More generally, if (W,S) is an even Coxeter system, we give a combinatorial condition on the diagram for (W,S) that determines whether or not two even systems for W have the same set of reflections. If (W,S) is even and (W,S) is not even, then these systems do not have the same set of reflections. A Coxeter group is said to be reflection independent if any two Coxeter systems (W,S) and (W,S) have the same set of reflections. We classify all reflection independent even Coxeter groups.Mathematics Subject Classifications (2000). 20F05, 20F55, 20F65, 51F15.     

A geometric characterization of Coxeter matroids

Coxeter matroids, introduced by Gelfand and Serganova, are combinatorial structures associated with any finite Coxeter group and its parabolic subgroup they include ordinary matroids as a specia case. A basic result in the subject is a geometric characterization of Coxeter matroids first stated by Gelfand and Serganova. This paper presents a self-contained, simple proof of a more general version of this geometric characterization.     

On the Maximally Clustered Elements of Coxeter Groups

We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements introduced by Losonczy and the author, which in turn constitute a superset of the i ji-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely, those Coxeter groups having finitely many maximally clustered elements. Remarkably, any simply laced Coxeter group having finitely many i ji-avoiding elements also turns out to be MC-finite.     

On Centralizers of Reflections in Coxeter Groups

We show that the centralizer of a reflection in a Coxeter groupis the semidirect product of a Coxeter group by a free group     

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.