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Sur un théorème de V. V. Deodhar et de M. J. Dyer sur les groupes de Coxeter

Jean-Yves Hée

Vol. 20 (2023), No. 2-3, 295–301
Abstract

Dans un groupe de Coxeter, tout sous-groupe engendré par des réflexions est aussi un groupe de Coxeter. Ce théorème a été démontré il y a un peu plus de trente ans indépendamment par V. V. Deodhar et par M. Dyer. Nous en donnons une autre démonstration, fondée sur la propriété suivante. Soit W un groupe engendré par un ensemble R d’éléments d’ordre 2. Alors, il existe une partie S de R telle que (W,S) soit un système de Coxeter dont R est l’ensemble des réflexions si et seulement s’il existe une opération de W sur un ensemble non vide E et une famille (𝒫r)rR de partitions de E remplissant certaines conditions simples de nature géométrique.

In a Coxeter group, any subgroup generated by reflections is also a Coxeter group. This theorem was proved a little more than thirty years ago independently by V. V. Deodhar and by M. Dyer. We give an alternative proof of this result that is based on the following property. Let W be a group generated by a set R of elements of order 2. Then, there exists a subset S of R such that (W,S) is a Coxeter system for which R is the set of reflections if and only if there exists an action of W on a nonempty set E and a family (𝒫r)rR of partitions of E satisfying some simple conditions of a geometrical nature.

À la mémoire de Jacques Tits

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Keywords
Coxeter systems, reflection subgroups, families of partitions, walls
Mathematical Subject Classification
Primary: 20F55
Milestones
Received: 10 December 2022
Revised: 26 April 2023
Accepted: 13 May 2023
Published: 13 September 2023
Authors
Jean-Yves Hée
LAMFA, UMR CNRS 7352
Université de Picardie Jules Verne
Amiens
France