A note on non-reduced reflection factorizations of Coxeter elements

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.Comment: Comments welcom

Hurwitz transitivity in elliptic Weyl groups and weighted projective lines

We continue the combinatorial description of thick subcategories in hereditary categories started by Igusa-Schiffler-Thomas and Krause. We show that for a weighted projective line $\mathbb{X}$ there exists an order preserving bijection between the thick subcategories of $\text{coh}(\mathbb{X})$ generated by an exceptional sequence and the factorization poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced generalized root system if the Hurwitz action is transitive on the reduced factorizations of $c$. By using combinatorial and group theoretical tools we show that this assumption on the transitivity of the Hurwitz action is fulfilled for a weighted projective line $\mathbb{X}$ of tubular type. In this case the factorization poset is given as the set of prefixes of a Coxeter transformation in certain elliptic Weyl groups described by Saito. As a byproduct we obtain a result on the Hurwitz action on non-reduced reflection factorizations in finite Coxeter groups which partially generalizes a result of Lewis-Reiner.Comment: Revision of the first version: corrected statement of Theorem 1.1, partially changed structure of the paper, minor corrections and change

Hurwitz action in Coxeter groups and extended Weyl groups with application in representation theory of finite dimensional algebras

Yahiatene S. Hurwitz action in Coxeter groups and extended Weyl groups with application in representation theory of finite dimensional algebras. Bielefeld: Universität Bielefeld; 2020

Reflection factorizations and quasi-Coxeter elements

Wegener P, Yahiatene S. Reflection factorizations and quasi-Coxeter elements. Journal of Combinatorial Algebra. 2023;7(1):127-157.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 reflec-tion 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 para-bolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups