Abstract. Let (W, S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W -action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W , i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.
Abstract. In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W also lie in this parabolic subgroup.
Abstract. We give complete classifications of the reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case-free proof is given of the well-known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where we give two distinct 'on the nose' classifications of reflection subgroups, as well as reproving Coxeter's conjecture on the isomorphism classes of reflection groups which occur. Geometric and combinatorial aspects of root systems play an essential role. Parameter sets for various types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.