To any complex reflection group W ⊂ GL(V ), one may attach a braid group B(W ), defined as the fundamental group of the space of regular orbits for the action of W on V [Broué et al. 98]. The “ordinary” braid group on n strings, introduced by [Artin 47], corresponds to the case of the symmetric group Sn, in its monomial reflection representation in GLn(C). More generally, any Coxeter group can ...

In recent work ([KW1],[KW2]), Kostant and Wallach construct an action of a simply connected Lie group A ≃ C( n 2 ) on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In [KW1], the authors show that A-orbits of dimension ( n 2 ) form Lagrangian submanifolds of regular adjoint orbits in gl(n). They describe ...

A polytope is called a Coxeter polytope if its dihedral angles are integer parts of π. In this paper we prove that if a noncompact Coxeter polytope of finite volume in IH has exactly n+3 facets then n ≤ 16. We also find an example in IH and show that it is unique. 1. Consider a convex polytope P in n-dimensional hyperbolic space IH. A polytope is called a Coxeter polytope if its dihedral angles...

In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph...

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. ...

An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups ...

We demonstrate that Coxeter groups allow for complex PT -symmetric deformations across the boundaries of all Weyl chambers. We compute the explicit deformations for the A2 and G2-Coxeter group and apply these constructions to Calogero-MoserSutherland models invariant under the extended Coxeter groups. The eigenspecta for the deformed models are real and contain the spectra of the undeformed cas...

Let 8 be an irreducible crystallographic root system in a Euclidean space V , with 8+ the set of positive roots. For α ∈8, k ∈Z, let H(α, k) be the hyperplane {v ∈ V : 〈α, v〉 = k}. We define a set of hyperplanes H = {H(δ, 1) : δ ∈ 8+} ∪ {H(δ, 0) : δ ∈ 8+}. This hyperplane arrangement is significant in the study of the affine Weyl groups. In this paper it is shown that the Poincaré polynomial of...

(1) The Poincar e polynomials of the nite irreducible Coxeter groups and the Poincar e series of the aane Coxeter groups on three generators are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. (2) Non-recursive methods for the computation of`standard reduced words' for (signed) permutations are described. The algebraic basis for both (1) and (2...