Archiv der Mathematik

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

A note on Weyl groups and root lattices

  • Barbara BaumeisterEmail author
  • Patrick Wegener


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.


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

Mathematics Subject Classification



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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].


  1. 1.
    Balnojan, S., Hertling, C.: Reduced and nonreduced presentations of Weyl group elements. arXiv:1604.07967 (2016)
  2. 2.
    Baumeister, B., Gobet, T., Roberts, K., Wegener, P.: On the Hurwitz action in finite Coxeter groups. J. Group Theory 20, 103–131 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bessis, D.: The dual braid monoid. Annales scientifiques de l’École Normale Supérieure 36(5), 647–683 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourbaki, N.: Lie Groups and Lie Algebras, Chapters 4–6. Springer, Berlin (2002)CrossRefGoogle Scholar
  5. 5.
    Brady, N., McCammond, J.P., Mühlherr, B., Neumann, W.D.: Rigidity of Coxeter groups and Artin groups. Geom. Dedicata 94(1), 91–109 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brady, T., Watt, C.: \(K(\pi , 1)^{\prime }\)s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa 2000), Geom. Dedicata, vol. 94, pp. 225–250 (2002)Google Scholar
  7. 7.
    Carter, R.W.: Conjugacy classes in the Weyl group. Compos. Math. 25, 1–59 (1972)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Caprace, P.E., Przytycki, P.: Twist-rigid Coxeter groups. Geom. Topol. 14, 2243–2275 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dörner, A.: Isotropieuntergruppen der artinschen Zopfgruppen. Bonner Mathematische Schriften 255, Univ. Bonn, Bonn (1993)Google Scholar
  10. 10.
    Dyer, M.J.: Reflection subgroups of Coxeter systems. J. Algebra 135(1), 57–73 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dyer, M.J., Lehrer, G.I.: Reflection subgroups of finite and affine Weyl groups. Trans. Am. Math. Soc. 363(11), 5971–6005 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang, J., Przytycki, P.: A step towards Twist Conjecture. arXiv:1708.00960 (2017)
  13. 13.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  14. 14.
    Mühlherr, B., Nuida, K.: Intrinsic reflections in Coxeter systems. J. Combin. Theory Ser. A 144, 326–360 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mühlherr, B.: The isomorphism problem for Coexter groups. arXiv:math.GR/0506572 (2005)
  16. 16.
    Taylor, D.: Reflection subgroups of finite complex reflection subgroups. J. Algebra 366, 218–234 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Vinberg, E.: Personal Communication (2017)Google Scholar
  18. 18.
    Wegener, P.: On the Hurwitz action in affine Coxeter groups. arXiv:1710.06694 (2017)

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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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