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Archiv der Mathematik

, Volume 111, Issue 5, pp 469–477 | Cite as

A note on Weyl groups and root lattices

  • Barbara Baumeister
  • Patrick Wegener
Article
  • 37 Downloads

Abstract

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 (WS) is a Coxeter system of finite rank n with set of reflections T and if \(t_1, \ldots t_n \in T\) are reflections in W that generate W, then \(P:= \langle t_1, \ldots t_{n-1}\rangle \) is a parabolic subgroup of (WS) of rank \(n-1\) (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (WS) is crystallographic as well, then all the reflections \(t \in T\) such that \(\langle P, t\rangle = W\) form a single orbit under conjugation by P.

Keywords

Weyl group Finite crystallographic Coxeter systems Finite crystallographic root lattices Dual Coxeter system Quasi-Coxeter elements Generation of dual Coxeter systems 

Mathematics Subject Classification

20F55 

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Notes

Acknowledgements

We like to thank Professor Ernest Vinberg for fruitful discussions with him as well as for explaining to us his idea of a uniform proof of Theorem 1.5. We also wish to thank the anonymous referee for helpful comments, and for making us aware of [16].

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universität BielefeldBielefeldGermany
  2. 2.Technische Universität KaiserslauternKaiserslauternGermany

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