We consider a cocompact discrete reflection group W of a CAT(0) space X . Then W becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex Σ(W, S) and the CAT(0) space X , and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group W .

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a certain geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examp...

Frieze patterns of numbers, introduced in the early 1970s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present survey aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter npolytopes with n + 3 facets, 4 ≤ n ≤ 7. Combined with results of Esselmann this gives the classification of all compact hyperbolic Coxeter n-polytopes with n + 3 facets, n ≥ 4. Polytopes in dimensions 2 and 3 were classified by Poincaré and And...

For W a finite Coxeter group, a formula is found for the size of W equivalence classes of subsets of a base. The proof is a case-by-case analysis using results and tables of Orlik and Solomon. As a corollary we obtain an alternating sum identity which generalizes a well-known identity from the theory of Coxeter groups.

We derive presentations of the interval groups related to all quasi-Coxeter elements in Coxeter group type D n . Type is only infinite family finite that admits proper elements. The we obtain are over a set generators bijection with what call Carter generating set, and relations those defined by diagram together twisted cycle or commutator relator, depending on whether element not. proof based ...

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by squares longest elements over connected subgraphs, and we prove that only relations either commutators or certain palindromic length 5 box relations. is motivated twist functors in algebraic geometry, but proof entirely Coxeter-theoretic. We also analogous set does not generate for all...

Generally we use definitions and notation as in [D]. In particular, (W,S) is a finitely generated Coxeter system, C is a building with Weyl group W , |C| is the Davis realization of C. We will, however, confuse the Coxeter group and its abstract Coxeter complex, denoting both by W ; in particular, |W | denotes the Davis complex. The W -valued distance in C will be denoted δC , while δ will be t...

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar [16] has given a framework for computing the Kazhdan-Lusztig polynomials which generally invo...

This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy conjecture of Propp that for the action of a “Coxeter element” of vertex toggles, the difference of indicator functions of symmetrically-located vertices is...