Related papers: Reflection length in non-affine Coxeter groups

On etudie divers aspects d'une formule qui compte les reflexions pleines dans les groupes de Coxeter finis. ***** We study several points about a formula which counts reflexions in a finite Coxeter group whose reduced decompositions involve…

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their…

We show that the Coxeter polytopes that have finite volume in their Vinberg domains are exactly the quasiperfect Coxeter polytopes of negative type, i.e. the Coxeter polytopes that are contained in their properly convex Vinberg domain, at…

In this paper we study affine reflection subgroups in arbitrary infinite Coxeter groups of finite rank. In particular, we study the distribution of roots of Coxeter groups in the root subsystems associated with affine reflection subgroups.…

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we tackle different topics around the field of definition of a hyperbolic…

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form X_n={x_0,x_1,...,x_n}, which is a subset of the standard infinite generating set for F.…

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group $W$, every pair $t,t'$ of distinct reflections lie in a unique maximal dihedral reflection subgroup of $W$. Our proof only relies on…

We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional…

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize…

For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to…

A finite subgroup of $GL(n,\mathbb C)$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all…

Brink and Howlett have introduced a partial ordering, called dominance, on the positive roots in the Tits realization of Coxeter groups (Math. Ann. 296 (1993), 179--190). Recently a concept called $\infty$-height is introduced to each…

In this article, we study two combinatorial problems concerning the set of reflections of a Coxeter system. The first problem asks whether the language of palindromic reduced words for reflections is regular, and the second is about finding…

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in…

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n <…

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a…

We give a streamlined proof of a theorem of Foxby and Halvorsen. The theorem states that certain relative K-groups made from chain complexes with bounded (but arbitrarily long) length coincide with similar K-groups in which one sets an…

We prove that among all right-angled Coxeter groups in hyperbolic 3-space, the group generated by reflections in the faces of a right-angled triangular bipyramid with three ideal and two finite vertices has the smallest covolume. The group…

We use probabilistic methods to prove that many Coxeter groups are incoherent. In particular, this holds for Coxeter groups of uniform exponent > 2 with sufficiently many generators.