Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T01:20:39.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetric presentations of Coxeter groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  20 June 2011

Ben Fairbairn
Ben Fairbairn
Affiliation:
Departamento de Matemáticas, Universidad de Los Andes, Carrera 1 No. 18A-12, Bogotá, Colombia
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.

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is, the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.

Keywords

symmetric generationCoxeter group

MSC classification

Secondary: 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 669 - 683
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Bray, J. N. and Curtis, R. T., Double coset enumeration of symmetrically generated groups, J. Group Theory 7 (2004), 167–185.Google Scholar
2.Bray, J. N. and Curtis, R. T., The Leech lattice, Λ and the Conway group ·0 revisited, Trans. Am. Math. Soc. 362 (2010), 1351–1369.Google Scholar
3.Bray, J. N., Curtis, R. T. and Hammas, A. M. A., A systematic approach to symmetric presentations, I, Involutory generators, Math. Proc. Camb. Phil. Soc. 119 (1996), 23–34.Google Scholar
4.Bray, J. N., Curtis, R. T., Parker, C. W. and Wierdorn, C. B., Symmetric presentations for the Fischer groups, I, The classical groups Sp6(2), Sp8(2), and 3O7(3), J. Alg. 265 (2003), 171–199.Google Scholar
5.Cannon, J. J. and the Magma Development Team, The Magma programming language (various versions up to Version 2.8), School of Mathematics and Statistics, University of Sydney (1993–2001).Google Scholar
6.Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., An atlas of finite groups (Oxford University Press, 1985).Google Scholar
7.Coxeter, H. S. M., Discrete groups generated by reflections, Annals Math. 35 (1934), 588–621.Google Scholar
8.Curtis, R. T., A survey of symmetric generation of sporadic simple groups, in The atlas of finite groups: ten years on, London Mathematical Society Lecture Note Series, Volume 249, pp. 39–57 (Cambridge University Press, 1998).Google Scholar
9.Curtis, R. T., Symmetric generation and existence of the Janko group J1, J. Group Theory 2 (1999), 355–366.Google Scholar
10.Curtis, R. T., Symmetric generation of groups with applications to many of the sporadic finite simple groups, Encyclopedia of Mathematics and Its Applications, Volume 111 (Cambridge University Press, 2007).Google Scholar
11.Curtis, R. T. and Fairbairn, B. T., Symmetric representation of the elements of the Conway group ·0, J. Symb. Computat. 44 (2009), 1044–1067.CrossRefGoogle Scholar
12.Fairbairn, B. T., On the symmetric generation of finite groups, PhD thesis, University of Birmingham (2009).Google Scholar
13.Fairbairn, B. T., Recent progress in the symmetric generation of groups, Proc. Conf. Groups St Andrews 2009, in press.Google Scholar
14.Fairbairn, B. T. and Müller, J., Symmetric generation of Coxeter groups, Arch. Math. 93 (2009), 1–10.Google Scholar
15.Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 1997).Google Scholar
16.Jansen, C., Lux, K., Parker, R. and Wilson, R., An atlas of Brauer characters (Oxford University Press, 1995).Google Scholar