Interval groups related to finite Coxeter groups. I
type: Item
instance of: scholarly article
title: Interval groups related to finite Coxeter groups. I
zbMATH Open document ID: 1515.20191
arXiv ID: 2103.06570
author: Barbara Baumeister
author: Georges Neaime
author: Sarah Rees
published in: Algebraic Combinatorics
publication date: 2023-05-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/2103.06570
review text: In the 1960s, \textit{F. Garside} introduced a new algebraic aspect to the study of the braid group \(B_{n}\) in his solution to the conjugacy problem in \(B_{n}\) [Q. J. Math., Oxf. II. Ser. 20, 235--254 (1969; Zbl 0194.03303)]: Garside defined a submonoid \(B^{+}_{n}\) of \(B_{n}\) and a distinguished element \(\Delta_{n} \in B^{+}_{n}\) with the property that every element \(g\in B_{n}\) can be written in the form \(g=\Delta_{n}^{m}g'\) with \(m \in \mathbb{Z}\) and \(g' \in B_{n}^{+}\). It was then realised that Garside's approach extends to all Artin groups of spherical type, independently in [\textit{E. Brieskorn} and \textit{K. Saito}, Invent. Math. 17, 245--271 (1972; Zbl 0243.20037)] and [\textit{P. Deligne}, Invent. Math. 17, 273--302 (1972; Zbl 0238.20034)]. In this paper, the authors derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type \(D_{n}\). Type \(D_{n}\) is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations that they obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted cycle or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements.
reviewed by: Egle Bettio
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 7681942
zbMATH Keywords: Coxeter group
zbMATH Keywords: Garside structures
zbMATH Keywords: quasi-Coxeter element
zbMATH Keywords: generalised non-crossing partitions
zbMATH Keywords: dual approach to Coxeter and Artin groups
zbMATH Keywords: interval (Garside) structures
zbMATH Keywords: Carter diagram
zbMATH Keywords: Artin(-Tits) group
MaRDI profile type: Publication
OpenAlex ID: W3133786132
cites work: Reflection group presentations arising from cluster algebras
cites work: A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements.
cites work: On the Hurwitz action in finite Coxeter groups
cites work: A note on Weyl groups and root lattices
cites work: The dual braid monoid
cites work: Springer theory in braid groups and the Birman-Ko-Lee monoid.
cites work: A new approach to the word and conjugacy problems in the braid groups
cites work: Factoring Euclidean isometries
cites work: Artin-Gruppen und Coxeter-Gruppen
cites work: Signed graphs, root lattices, and Coxeter groups
cites work: Q5668957
cites work: Bestvina's normal form complex and the homology of Garside groups.
cites work: Foundations of Garside theory
cites work: Les immeubles des groupes de tresses généralises
cites work: Reflection subgroups of Coxeter systems
cites work: On minimal lengths of expressions of Coxeter group elements as products of reflections
cites work: THE BRAID GROUP AND OTHER GROUPS
cites work: Braid groups and quiver mutation
cites work: Artin group presentations arising from cluster algebras
cites work: Artin groups of Euclidean type
cites work: Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)
cites work: Finite Unitary Reflection Groups
cites work: Formula for the reflection length of elements in the group \(G(m,p,n)\).
DOI: 10.5802/ALCO.266
description: scientific article; zbMATH DE number 7681942