Hostname: page-component-5cf477f64f-tx7qf Total loading time: 0 Render date: 2025-04-02T11:58:09.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Special involutions and bulky parabolic subgroups in finite Coxeter groups

Part of: Special aspects of infinite or finite groups Ordered sets

Published online by Cambridge University Press:  09 April 2009

Götz Pfeiffer  and
Gerhard Röhrle
Götz Pfeiffer
Affiliation:
Department of MathematicsNational University of IrelandGalwayIreland e-mail: [email protected]
Gerhard Röhrle
Affiliation:
School of MathematicsUniversity of BirminghamBirmingham B15 2TTUnited Kingdom e-mail: [email protected]
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.

The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 06A07: Combinatorics of partially ordered sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 1 , August 2005 , pp. 141 - 147
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bourbaki, N., Groupes et algèbres de Lie. Chapitres IV–VI (Hermann, Paris, 1968).Google Scholar
[2]Brink, B. and Howlett, R. B., ‘Normalizers of parabolic subgroups in Coxeter groups’, Invent. Math. 136 (1999), 323–351.CrossRefGoogle Scholar
[3]Felder, G. and Veselov, A. P., ‘Coxeter group actions on the complement of hyperplanes and special involutions’, J. Eur. Math. Soc. (JEMS) 7 (2005), 101–116.CrossRefGoogle Scholar
[4]Fleischmann, P. and Janiszczak, I., ‘The lattices and Möbius functions of stable closed subrootsystems and hyperplane complements for classical Weyl groups’, Manuscripta Math. 72 (1991), 375–403.CrossRefGoogle Scholar
[5]Fleischmann, P. and Janiszczak, I., ‘Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups’, J. Combin. Theory Ser A 63 (1993), 257–274.CrossRefGoogle Scholar
[6]Geck, M. and Pfeiffer, G., Characters offinite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monogr. New Series 21 (Oxford University Press, New York, 2000).CrossRefGoogle Scholar
[7]Howlett, R. B., ‘Normalizers of parabolic subgroups of reflection groups’, J. London Math. Soc. (2) 21 (1980), 62–80.CrossRefGoogle Scholar
[8]Lehrer, G. I., ‘On hyperoctahedral hyperplane complements’, in: The Arcata Conference on Representations of Finite Groups (Arcata, CA, 1986), Proc. Sympos. Pure Math. 47 (Amer. Math. Soc., Providence, RI, 1987) pp. 219–234.CrossRefGoogle Scholar
[9]Lehrer, G. I., ‘On the Poincaré series associated with Coxeter group actions on complements of hyperplanes’, J. London Math. Soc. (2) 36 (1987), 275–294.CrossRefGoogle Scholar
[10]Richardson, R. W., ‘Conjugacy classes of involutions in Coxeter groups’, Bull. Austral. Math. Soc. 26 (1982), 1–15.CrossRefGoogle Scholar
[11]Springer, T. A., ‘Some remarks on involutions in Coxeter groups’, Comm. Algebra 10 (1982), 631–636.CrossRefGoogle Scholar