Geodesic
Interval groups related to finite Coxeter groups I
Baumeister, Barbara1 ; Neaime, Georges2 ; Rees, Sarah3
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, Tome 6 (2023) no. 2, pp. 471-506

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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.

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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.

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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Interval groups related to  finite {Coxeter} groups {I}},
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Baumeister, Barbara; Neaime, Georges; Rees, Sarah. Interval groups related to  finite Coxeter groups I. Algebraic Combinatorics, Tome 6 (2023) no. 2, pp. 471-506. doi: 10.5802/alco.266

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