We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.

No abstract

We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis (2003) and the question whether there is an analogue of the well-known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis–Reiner as well as Baumeister–Gobet–Roberts and the first author on the Hurwitz action in finite and affine Coxeter groups. Further we show that in an arbitrary Coxeter group all reduced reflection factorizations of the same element appear in the same Hurwitz orbit after a suitable extension by simple reflections. As parabolic quasi-Coxeter elements play an outstanding role in the study of the Hurwitz action, we aim to characterize these elements. We provide a characterization of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.

In this paper, which is a continuation of [3], we study extended Weyl groups of domestic and wild type. We start with an extended Coxeter-Dynkin diagram, attach to it a set of roots (vectors) in an R-space and define the extended Weyl groups as groups that are generated by the reflections related to these roots, the so called simple reflections. We relate to such a group W a generalized root system, and determine the structure of W . In particular we present a normal form for the elements of W . Further we define Coxeter transformations in W , and show the transitivity of the Hurwitz action on the set of reduced reflection factorizations of a Coxeter transformation where the reflections are the conjugates of the simple reflections in W (see Theorem 1.1).We give an application of Theorem 1.1 and of the results in [3] in the representation theory of hereditary categories by establishing a bijection between the poset of certain thick subcategories of a hereditary category, namely the category coh(X) of coherent sheaves on a weighted projective line X, and the poset of elements in W that are below a Coxeter transformation in terms of a fixed partial order (see Theorem 1.4). Contents1. Introduction Structure of the paper Acknowledgments 2. Notation, terminology and basic facts 2.1. The extended Coxeter-Dynkin diagram 2.2. The extended space 2.3. The extended Weyl group 2.4. The Hurwitz action 3. Normal form and translation vector in extended Weyl groups 3.1. The extended root system and its reflections 3.2. The structure of W 3.3. The normal form for elements in W 4. The Coxeter transformations 4.1. The conjugacy of Coxeter transformations 4.2. Reflection length of Coxeter transformations 5. A remark on Coxeter groups whose diagram is a star 6. Hurwitz transitivity for Coxeter transformations of extended Weyl groups of domestic or wild type 6.1. Hurwitz action in Coxeter groups 6.2. The Proof of Theorem 1.1 7. Application of the main theorem

No abstract