Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-08T08:54:44.346Z Has data issue: false hasContentIssue false
  • English
  • Français

ON PAIRED ROOT SYSTEMS OF COXETER GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  18 June 2018

XIANG FU
XIANG FU*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing, China email fuxiang@math.pku.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper examines a systematic method of constructing a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of nonstandard geometric representations. This method can be employed to construct generalizations of root systems for a large family of linear groups generated by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

Keywords

Coxeter groupsKac–Moody Lie algebrasroot systemsTits cone

MSC classification

Primary: 20F55: Reflection and Coxeter groups
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata 20F65: Geometric group theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 3 , December 2018 , pp. 347 - 365
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Björner, A. and Brenti, F., Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, GTM, 231 (Springer, New York, 2005).Google Scholar
Bourbaki, N., Groupes et algebras de Lie, Chapitres 4, 5 et 6 (Hermann, Paris, 1968).Google Scholar
Caprace, P. E. and Rémy, B., ‘Groups with a root group datum’, Innov. Incidence Geom. 9 (2009), 5–77.Google Scholar
Casselman, W. A., ‘Computation in Coxeter groups I. Multiplication’, Electron. J. Combin. 9(1) (2002), Research Paper 25, 22 pp. (electronic).Google Scholar
Casselman, W. A., ‘Computation in Coxeter groups II. Constructing minimal roots’, Represent. Theory 12 (2008), 260–293.Google Scholar
Deodhar, V., ‘On the root system of a Coxeter group’, Comm. Algebra 10(6) (1982), 611–630.Google Scholar
Dyer, M., ‘Hecke algebras and reflections in Coxeter groups’, PhD Thesis, University of Sydney, 1987.Google Scholar
Dyer, M., ‘Reflection subgroups of Coxeter systems’, J. Algebra 135 (1990), 57–73.Google Scholar
Dyer, M., On rigidity of abstract root systems of Coxeter groups, arXiv:1011.2270 [math.GR], Preprint, 2010.Google Scholar
Fu, X., ‘Root systems and reflection representations of Coxeter groups’, PhD Thesis, University of Sydney, 2010.Google Scholar
Fu, X., ‘Non-orthogonal geometric realizations of Coxeter groups’, J. Aust. Math. Soc. 97(2) (2014), 180–211.Google Scholar
Hohlweg, C., Labbé, J. P. and Ripoll, V., ‘Asymptotical behaviour of roots of infinite Coxeter groups’, Canad. J. Math. 66(2) (2014), 323–353.Google Scholar
Humphreys, J., Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., 29 (Cambridge University Press, Paris, 1990).Google Scholar
Moody, R. V. and Pianzola, A., ‘On infinite root systems’, Trans. Amer. Math. Soc. 315 (1989), 661–696.Google Scholar