Related papers: Absolute order and involutions

In this note, we characterize affine and non-affine Coxeter systems among all Coxeter systems in terms of the structure of their reflection orders. For an infinite irreducible system $(W,S)$, we show that affineness can be characterized in…

We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new…

Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then B acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A=A(W) of W. In this paper, we show…

Suppose that $W$ is a finite Coxeter group and $W_J$ a standard parabolic subgroup of $W$. The main result proved here is that for any for any $w \in W$ and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon, where $m =…

We present a complete classification for the epimorphisms from the affine Artin-Tits groups $A(\widetilde{A}_{n})$ to the corresponding Coxeter groups $W(\widetilde{A}_{n})$, for $n\geq 1$.

A group $G$ is said to be just infinite if $G$ itself is infinite but all proper quotients of $G$ are finite. We show that a Coxeter group $W_\Gamma$ is just infinite if and only if $\Gamma$ is isomorphic to one of the following graphs:…

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria…

The shard intersection order is a new lattice structure on a finite Coxeter group W which encodes the geometry of the reflection arrangement and the lattice theory of the weak order. In the case where W is the symmetric group, we…

The absolute order on the hyperoctahedral group $B_n$ is investigated. It is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed. Moreover,…

The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by problems in Kazhdan--Lusztig theory, Matthew Dyer introduced…

Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s \in S$. If there exists $t\in S$ such that $s$ and $t$ do not…

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…

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of…

We study the sublattices of the rank $2^d$ Barnes-Wall lattices $\bw d$ which occur as fixed points of involutions. They have ranks $2^{d-1}$ (for dirty involutions) or $2^{d-1}\pm 2^{k-1}$ (for clean involutions), where $k$, the defect, is…

Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter…

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let…

Let $(W, R)$ be a Coxeter system and let $w \in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w = uv$ for some $v$ in $W$ such that $\ell(w) = \ell(u) +…

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on…

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…

Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the…