The hyperbolic cover of an elliptic Weyl group
Barbara Baumeister, Patrick Wegener
TL;DR
The paper addresses the lack of Hurwitz transitivity for reduced reflection factorizations in tubular elliptic Weyl groups by passing to hyperbolic covers tilde{W} and hat{W}, which are shown to be isomorphic extended Coxeter systems of star type. It develops a detailed construction of the hyperbolic extended spaces and the Eichler-Siegel framework, yielding a normal form and a clear semidirect product structure for tilde{W}. This machinery enables a uniform proof of Hurwitz transitivity for reduced factorizations of the Coxeter transformation tilde{c} and transfers orbit information to the elliptic setting, culminating in an application to weighted projective lines of tubular type: an order-preserving bijection between thick subcategories generated by exceptional sequences in coh(X) and the interval [id, tilde{c}] in the absolute order. The appendix shows that hyperbolic covers of Coxeter systems do not introduce new groups, reinforcing the foundational role of hyperbolic extensions in this framework.
Abstract
In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For $\tilde{c}$ a Coxeter transformation in $\tilde{W}$ we can conclude the Hurwitz transitivity of the braid group action on the set of reduced reflection factorizations of $\tilde{c}$ from the Hurwitz transitivity in extended Coxeter systems of star type. This then enables us to establish for a weighted projective line $\mathbb{X}$ of tubular type an order preserving bijection between the poset of thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and the poset $[\mathrm{id}, \tilde{c}]$ ordered by the absolute order. In an Appendix, we study the hyperbolic cover of a Coxeter system.