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Patrick Wegener

Publications and source records attributed to Patrick Wegener.

12 recordsLinked to original sources
arXiv2026

The Hurwitz Action in the Affine Symmetric Group

Let $W$ be an affine Coxeter group of type $\widetilde{A}_n$, that is, the affine symmetric group $\widetilde{S}_N$ with $N=n+1$, let $T$ be its set of reflections, and let $\mathrm{Red}_T(w)$ be the set of reduced reflection factorizations of an element $w\in W$. The braid group acts on $\mathrm{Red}_T(w)$ by the Hurwitz action. For finite Coxeter groups it is known exactly when this action is transitive, namely precisely for the parabolic quasi-Coxeter elements. We address this problem for the affine type $\widetilde{A}_n$ by determining all orbits of $\mathrm{Red}_T(w)$.

math.GR↗
arXiv2024

The hyperbolic cover of an elliptic Weyl group

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.

math.GR↗
arXiv2021

Reflection factorizations and quasi-Coxeter elements

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 reflection 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 and the question whether there is an anlogue 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 and Reiner as well as Baumeister, Gobet, Roberts and the first author on the Hurwitz in finite 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 give characterizations of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.

math.GR↗
arXiv2021

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

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.

math.RT↗
arXiv2019

Reflection groups and quiver mutation: Diagrammatics

We extend Carter's notion of admissible diagrams and attach a "Dynkin-like" diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. Remarkably, such a diagram turns out to be cyclically orientable if and only if it is isomorphic to the underlying graph of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore we show that each diagram encodes a natural presentation of the Weyl group as reflection group. The latter one extends work of Cameron, Seidel and Tsaranov as well as Barot and Marsh.

math.CO↗
arXiv2018

Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case

We start the systematic study of extended Weyl groups, and continue the combinatorial description of thick subcategories in hereditary categories started by Ingalls-Thomas, 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 $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and a subposet of the interval poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced extended root system if the Hurwitz action is transitive on the reduced reflection factorizations of $c$ that generate the Weyl group. 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.

math.RT↗
arXiv2017

On the Hurwitz action in affine Coxeter groups

We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.

math.GR↗
arXiv2017

A note on Weyl groups and crystallographic root lattices

We follow the dual approach to Coxeter systems and show for Weyl groups a criterium which decides whether a set of reflections is generating the group depending on the root and the coroot lattice. Further we study special generating sets involving a parabolic subgroup and show that they are very tame.

math.GR↗
arXiv2015

On the Hurwitz action in finite Coxeter groups

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.

math.GR↗
arXiv2014

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.

math.GR↗