Related papers: Intrinsic reflections in Coxeter systems

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well…

The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of…

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…

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$, we define the Coxeter pop-tsack torsing operator $\mathsf{Pop}_T:W\to W$ by $\mathsf{Pop}_T(w)=w\cdot\pi_T(w)^{-1}$, where $\pi_T(w)$ is the join in the…

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. While these partial orders are of independent combinatorial interest, our investigation is motivated by a connection between the…

For a finite irreducible Coxeter group $(W,S)$ with a fixed Coxeter element $c$ and set of reflections $T$, Defant and Williams define a pop-tsack torsing operation $\mathrm{Popt}\colon W \to W$ given by $\mathrm{Popt}(w) = w \cdot…

We give explicit generators for ideals of two classes of subspace arrangements embedded in certain reflection arrangements, generalizing results of Li-Li and Kleitman-Lovasz. We also give minimal generators for the ideals of arrangements…

It is well known that a line in $\mathbb R^2$ is not a set of injectivity for the spherical means for odd functions about that line. We prove that any line passing through the origin is a set of injectivity for the twisted spherical means…

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…

Let $W$ be a finite Coxeter group with Coxeter generating set $S=\{s_1,\ldots,s_n\}$, and $\rho$ be a complex finite dimensional representation of $W$. The characteristic polynomial of $\rho$ is defined as \begin{equation*}…

Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid…

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of…

We prove that the centralizer of a Coxeter element in an irreducible Coxeter group is the cyclic group generated by that Coxeter element.

In this paper we show for the first time the phenomenon of negative reflection in a simple mechanical structure. The latter is a grating of fixed inclusions embedded in a linear elastic matrix. Numerical analyses for out-of-plane shear…

Let $(W, S)$ be a Coxeter system of rank $n$ and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) < \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper we will show that this still holds for $q =…

A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of…

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in [8] to construct Elnitsky tilings for any finite Coxeter group.…

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs and Gyoja proved that every irreducible representation of the Iwahori-Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an…

In this note, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous generator (left-or right-continuous). By a comparison theorem establish here for…