Reflection factorizations and quasi-Coxeter elements

authored by

Patrick Wegener, Sophiane Yahiatene

Abstract

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Bielefeld University

Type

Article

Journal

Journal of Combinatorial Algebra

Volume

7

Pages

127-157

No. of pages

31

ISSN

2415-6302

Publication date

25.05.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics, Algebra and Number Theory

Electronic version(s)

https://arxiv.org/abs/2110.14581 (Access: Open)
https://doi.org/10.4171/JCA/70 (Access: Open)