Contents Root systems and generalized associahedra 1 Root systems and generalized associahedra 3 Lecture 1. Reflections and roots 5 1.1. The pentagon recurrence 5 1.2. Reflection groups 6 1.3. Symmetries of regular polytopes 8 1.4. Root systems 11 1.5. Root systems of types A, B, C, and D 13 Lecture 2. Dynkin diagrams and Coxeter groups 15 2.1. Finite type classification 15 2.2. Coxeter groups ...

We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements of Losonczy and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely those Coxeter groups havin...

Given an involutive automorphism θ of a Coxeter system (W,S), let I(θ) ⊆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 all reduced S-expressions (also known as admissible sequences, reduced Iθ-expressions, or involution words) for any given w ∈ I(θ). This can be viewed as an analogue of the well-known word pro...

We investigate which Weyl groups have a Coxeter presentation and which of them at least have the presentation by conjugation with respect to their root system. For most concepts of root systems the Weyl group has both. In the context of extended affine root systems (EARS) there is a small subclass allowing a Coxeter presentation of the Weyl group and a larger subclass allowing the presentation ...

In this article, we consider infinite, non-affine Coxeter groups. These are known to be of exponential growth. We consider the subsets of minimal length coset representatives of parabolic subgroups and show that these sets also have exponential growth. This is achieved by constructing a reflection subgroup of our Coxeter group which is isomorphic to the universal Coxeter group on three generato...

Let A be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of A are isometric. We prove that A is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open questio...

Let C(T ) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY (T ) be a natural quotient of C(T ), and if C(T ) is simply-laced (which means all the relations between the generators has order 2 or 3), CY (T ) is a generalized Coxeter group, too . Let At,n be a group which contains t Abelian groups generated by n elements. The...

Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W . In this paper, we define a more general notion of Ω-sortable elements, where Ω is an arbitrary orientation of the diagram, and show that the key properties of c-sortable elements...

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d + 3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88–91] and the second author [...

In this paper, M denotes a Dynkin diagram defined over an index set I and (W, {ri}i∈I) will be the associated Coxeter system. The diagram M is called simply-laced if it has only single bonds. For any subset J ⊂ I, MJ denotes the subdiagram of M defined over J and WJ will be the subgroup of W having generator set {ri}i∈J . The pair (WJ , {ri}i∈J) then is a Coxeter system with diagram MJ . Let X ...