Extended Weyl groups, Hurwitz transitivity and weighted projective lines II: a uniform approach

TL;DR

This paper explores the structure of extended Weyl groups, introduces hyperbolic covers as extended Coxeter groups, and demonstrates Hurwitz transitivity, with applications in algebraic representation theory and singularity theory.

Contribution

It establishes the hyperbolic covers of extended Weyl groups as extended Coxeter groups of star type and proves Hurwitz transitivity in this context, linking algebraic and geometric theories.

Findings

Hyperbolic covers are extended Coxeter groups of star type.

Hurwitz action is transitive on reduced reflection factorizations.

Applications connect Weyl groups to algebraic and singularity theories.

Abstract

We continue the study of extended Weyl groups $W$, which are reflection groups. Further we recall the definition of a hyperbolic cover of an extended Weyl group, and show that the hyperbolic covers of the extended Weyl groups are extended Coxeter groups, which had been introduced by Looijenga and discussed by people from different mathematical areas. More precisely the hyperbolic covers are the extended Coxeter groups of star type. We define simple reflections and Coxeter transformations in these groups, and show the transitivity of the Hurwitz action on the set of reduced reflection factorizations of a Coxeter transformation in the extended Coxeter groups of star type $\mathcal{W}$, where the reflections are the conjugates of the simple reflections in $\mathcal{W}$. We give two applications of our results. In the context of representation theory of algebras, we establish an isomorphism between the poset of thick subcategories that are generated by exceptional sequences of a hereditary connected ext-finite abelian $k$-category with a tilting object, $k$ algebraically closed of characteristic $0$, and the poset of elements in the extended Weyl group that are below a Coxeter transformation with respect to the absolute order. The second application concerns the theory of unimodal singularities. In particular, we provide an answer to a question of Brieskorn for the classical monodromy operator in the case of hyperbolic singularities.