Related papers: Semidirect product decomposition of Coxeter groups

The largest finite subgroup of O(4) is the noncrystallographic Coxeter group $W(H_{4})$ of order 14400. Its derived subgroup is the largest finite subgroup $W(H_{4})/Z_{2}$ of SO(4) of order 7200. Moreover, up to conjugacy, it has five…

Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincar\'e series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly…

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…

Given an involutive automorphism $\theta$ of a Coxeter system $(W,S)$, let $\mathfrak{I}(\theta) \subseteq W$ denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect…

In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For…

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_\Gamma$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_\Gamma)$ virtually surjects onto some infinite Coxeter group. In particular,…

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…

The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an…

Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence L(1), L(2), ..., L(n) of graph of groups decompositions of G with edge groups in C such that L(1) is the trivial…

In this paper, we study affine Deligne--Lusztig varieties $X_w(b)$ when the finite part of the element $w$ in the Iwahori--Weyl group is a partial $\sigma$-Coxeter element. We show that such $w$ is a cordial element and $X_w(b) \neq…

We develop new and precise geometric descriptions of the conjugacy class $[x]$ and coconjugation set $\operatorname{C}(x,x') = \{ y \in \overline{W} \mid yxy^{-1} = x' \}$ for all elements $x,x'$ of any affine Coxeter group $\overline{W}$.…

Let $(W,S)$ be a Coxeter system and write $P_W(q)$ for its Poincar\'e series. Lusztig has shown that the quotient $P_W(q^2)/P_W(q)$ is equal to a certain power series $L_{W}(q)$, defined by specializing one variable in the generating…

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations…

Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that…

Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $\phi$ of $W$ over the finite field $F_2$ of two elements. The action of $\phi(W)$ on $F_2^n$ by left multiplication is…

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how…

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…

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case)…

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(\Phi_{x}) = \Phi_{y}$ where $ w,x,y\in W$ and $ \Phi_{x}$, $\Phi_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a…

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group…