Related papers: Coxeter systems for which the Brink-Howlett automa…

We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular…

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…

We provide a proof for one version of Pisot conjecture. We make use of the weak mixing property of the subshift of finite type derived from the prefix-suffix automaton to conclude that the substitution dynamical system has pure discrete…

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the…

In this paper, we give a class of reflection rigid Coxeter systems. Let $(W,S)$ be a Coxeter system. Suppose that (1) for each $s,t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a maximal spherical subset of $S$, (2) there does not exist a…

Let $K$ be a finite extension of $\mathbb{Q}$ and $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{B}$ be a primitive collection of ideals in $\mathcal{O}_K$. We show that any $\mathfrak{B}$-free system is essentially minimal.…

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence…

This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms-which generalises and unifies the notion of bisimulation to the whole class of weighted automata-and the unicity of a…

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$…

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter…

We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with $(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular, the non-trivial isomorphisms fall into a small number of cases which are highly…

We construct models for the classifying spaces of coabelian subgroups of right-angled Coxeter groups as homotopy orbit spaces of real moment-angle complexes, generalizing well-known models for the classifying space of a right-angled Coxeter…

Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each…

Let $X$ be the Cantor set and $\phi$ be a minimal homeomorphism on $X\times\T$. We show that the crossed product $C^*$-algebra $C^*(X\times\T,\phi)$ is a simple $A\T$-algebra provided that the associated cocycle takes its values in…

For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={D_w}_{w\in W} such that each D_w contains as a direct summand (or is equal to) the indecomposable Soergel bimodule B_w. When…

Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial…

Given an arbitrary Coxeter system $(W,S)$ and a nonnegative integer $m$, the $m$-Shi arrangement of $(W,S)$ is a subarrangement of the Coxeter hyperplane arrangement of $(W,S)$. The classical Shi arrangement ($m=0$) was introduced in the…

The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top…

In this paper, we show that the center of every Coxeter group is finite and isomorphic to $(\Z_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where…

The purpose of this paper is to discuss the validity of the assumptions (W) and (S) stated in a previous work, about the torsion in the modular l-adic cohomology of Deligne-Lusztig varieties associated to Coxeter elements. We prove that…