Related papers: Involution Products in Coxeter Groups

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the…

It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathcal{Y}$. In this paper,…

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator,…

In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e.,…

This paper is a sequel to work of Dynkin on subroot lattices of root lattices and to work of Carter on presentations of Weyl group elements as products of reflections. The quotients $L/L_1$ are calculated for all irreducible root lattices…

We continue the study of extended Weyl groups $W$, which are reflection groups. Further we recall the definition of a hyperbolic cover of an extended Weyl group, and show that the hyperbolic covers of the extended Weyl groups are extended…

We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable…

If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative…

We revisit the structure of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, 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 Carter and Orlik/Solomon. As a corollary we obtain an alternating…

The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic…

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every…

To a Coxeter group $W$ one associates a quandle $X_W$ from which one constructs a group $\mathrm{Ad}(X_W)$. This group turns out to be an intermediate object between $W$ and the associated Artin group. By using a result of Akita, we prove…

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^\theta$, where $x$ is an element of $W$ and $W_L^\theta$ the subgroup of fixed points of an automorphism $\theta$ of order at most two of…

Orbits of the Weyl reflection groups attached to the simple Lie groups $A_2, C_2, G_2$ and Coxeter group $H_2$ are considered. For each of the groups products of any two orbits are decomposed into the union of the orbits. Results are…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main…

Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positive root system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let…