扩展Weyl群、Hurwitz传递性与加权射影直线 II：一种统一方法
Extended Weyl groups, Hurwitz transitivity and weighted projective lines II: a uniform approach
AI总结：
本文研究扩展Weyl群的双曲覆盖，证明其为星型扩展Coxeter群，并证明Hurwitz作用在Coxeter变换既约反射分解上的传递性，最后给出在代数表示论与单模奇点理论中的两个应用。
摘要：
我们继续研究作为反射群的扩展Weyl群 $W$。进一步，我们回顾扩展Weyl群的双曲覆盖的定义，并证明扩展Weyl群的双曲覆盖是扩展Coxeter群，后者由Looijenga引入并被不同数学领域的学者讨论。更准确地说，双曲覆盖是星型的扩展Coxeter群。我们在这些群中定义单反射和Coxeter变换，并证明在星型扩展Coxeter群 $\mathcal{W}$ 中，Hurwitz作用在Coxeter变换的既约反射分解集合上的传递性，其中反射是 $\mathcal{W}$ 中单反射的共轭。我们给出结果的两个应用。在代数表示论背景下，我们建立了一个偏序集同构，一方是具有倾斜对象的遗传连通ext-finite阿贝尔 $k$-范畴（$k$ 是特征 $0$ 的代数闭域）中由例外序列生成的厚子范畴构成的偏序集，另一方是扩展Weyl群中关于绝对序位于Coxeter变换之下的元素构成的偏序集。第二个应用涉及单模奇点理论。特别地，我们针对双曲奇点情形，为Brieskorn关于经典单值算子的问题提供了答案。
英文摘要：
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.
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