Interval groups related to  finite Coxeter groups I

Baumeister, Barbara; Neaime, Georges; Rees, Sarah

doi:10.5802/alco.266

Interval groups related to finite Coxeter groups I

Baumeister, Barbara ¹; Neaime, Georges ²; Rees, Sarah ³

¹ Fakultät für Mathematik Universität Bielefeld Postfach 10 01 31 33501 Bielefeld Germany
² Fakultät für Mathematik Universität Bielefeld Postfach 10 01 31 33615 Bielefeld Germany
³ School of Mathematics, Statistics and Physics University of Newcastle Newcastle NE1 7RU UK

Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 471-506.

Abstract

We 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 we 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.

In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations [4, 5]. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.

Received: 2022-02-06
Revised: 2022-08-26
Accepted: 2022-09-02
Published online: 2023-05-03

DOI: 10.5802/alco.266

Classification: 20F55, 20F36
Keywords: Coxeter groups, Quasi-Coxeter elements, Carter diagrams, Artin(–Tits) groups, dual approach to Coxeter and Artin groups, generalised non-crossing partitions, Garside structures, Interval (Garside) structures.

Author's affiliations:

Baumeister, Barbara ¹; Neaime, Georges ²; Rees, Sarah ³

¹ Fakultät für Mathematik Universität Bielefeld Postfach 10 01 31 33501 Bielefeld Germany
² Fakultät für Mathematik Universität Bielefeld Postfach 10 01 31 33615 Bielefeld Germany
³ School of Mathematics, Statistics and Physics University of Newcastle Newcastle NE1 7RU UK

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Interval groups related to  finite {Coxeter} groups {I}},
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     pages = {471--506},
     publisher = {The Combinatorics Consortium},
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Baumeister, Barbara; Neaime, Georges; Rees, Sarah. Interval groups related to  finite Coxeter groups I. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 471-506. doi : 10.5802/alco.266. https://alco.centre-mersenne.org/articles/10.5802/alco.266/

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