Extended Weyl groups, Hurwitz transitivity and weighted projective lines II: The wild case

Abstract

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