Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-26T22:47:01.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Descent Algebras

Published online by Cambridge University Press:  20 November 2018

Christophe Hohlweg
Christophe Hohlweg*
Affiliation:
The Fields Institute, 222 College Street, Toronto, ON, M5T 3J1 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.

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\mathbb{Z}$-module ${{\mathcal{D}}_{A}}\left( W \right)$ of the group algebra $\mathbb{Z}W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci–Reutenauer algebra.

Keywords

20F5505E15Coxeter groupSolomon descent algebradescent set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 535 - 546
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Atkinson, M. D., A new proof of a theorem of Solomon. Bull. London Math. Soc. 18(1986), no. 4, 351–354.Google Scholar
[2] Bergeron, F., Bergeron, N., Howlett, R. B., and Taylor, D. E., A decomposition of the descent algebra of a finite Coxeter group. J. Algebraic Combin. 1(1992), no. 1, 23–44.Google Scholar
[3] Bonnafé, C. and Hohlweg, C., Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups. Ann. Inst. Fourier (Grenoble) 56(2006), no. 1, 131–181.Google Scholar
[4] Bourbaki, N., Éléments de mathématique, Chap. 4-6, Actualités Scientifiques et Industrielles 1337, Hermann, Paris, 1968.Google Scholar
[5] Geck, M. and Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. London Mathematical Society Monographs 21, Oxford University Press, New York, 2000.Google Scholar
[6] Humphreys, J. E., Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990.Google Scholar
[7] Jőllenbeck, A., Nichtkommutative Charaktertheorie der symmetrischen Gruppen. Bayreuth. Math. Schr. 56(1999), 1–41.Google Scholar
[8] Mantaci, R. and Reutenauer, C., A generalization of Solomon's algebra for hyperoctahedral groups and other wreath products. Comm. Algebra 23(1995), no. 1, 27–56.Google Scholar
[9] Moszkowski, P., Généralisation d’une formule de Solomon relative à l’anneau de groupe d’un groupe de Coxeter. C. R. Acad. Sci. Paris Sér I Math. 309(1989), no. 8, 539–541.Google Scholar
[10] Reutenauer, C., Free Lie algebras, London Mathematical Society Monographs New Series 7(1993).Google Scholar
[11] Solomon, L., A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(1976), 255–264.Google Scholar
[12] Thibon, J.-Y., Lectures on noncommutative symmetric functions. In: Interaction of Combinatorics and Representation Theory. MSJ Mem. 11, Mathematical Society of Japan, Tokyo, 2001, pp. 39–94.Google Scholar
[13] Tits, J., Buildings of Spherical Type and Finite Bn-Pairs, Lecture Notes in Mathematics 386, Springer-Verlag, Berlin, 1974.Google Scholar