A note on Weyl groups and root lattices

We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if ( W ,  S ) is a Coxeter system of finite r...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Archiv der Mathematik 2018-11, Vol.111 (5), p.469-477
Hauptverfasser: Baumeister, Barbara, Wegener, Patrick
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if ( W ,  S ) is a Coxeter system of finite rank n with set of reflections T and if t 1 , … t n ∈ T are reflections in W that generate W , then P : = ⟨ t 1 , … t n - 1 ⟩ is a parabolic subgroup of ( W ,  S ) of rank n - 1 (Baumeister et al. in J Group Theory 20:103–131, 2017 , Theorem 1.5). Here we show if ( W ,  S ) is crystallographic as well, then all the reflections t ∈ T such that ⟨ P , t ⟩ = W form a single orbit under conjugation by P .
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-018-1234-5