Related papers: First-Order Aspects of Coxeter Groups

We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along…

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear…

By Theorem~1.12 of the paper "A Class of Representations of Hecke Algebras", if $W$ is a Coxeter group whose proper parabolic subgroups are finite, and if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a…

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the…

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing…

The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D.…

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)\in\{2\}\cup 4\N$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd,…

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion…

Popov classified crystallographic complex reflection groups by determining lattices they stabilize. These analogs of affine Weyl groups have infinite order and are generated by reflections about affine hyperplanes; most arise as the…

This paper and its sequel describe the irreducible representations of the rational Cherednik algebra $H_c(W)$ for a finite Coxeter group $W$ of type $H_4$, $F_4$ with equal parameters, $E_6$, $E_7$, and $E_8$, when $c$ is not a…

We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The…

For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph…

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2 (h) where h is the…

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is…

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the…

We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count…

Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits' bilinear form associated to the standard root system of…

Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As…

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework,…

Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we…