Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



On the generic degrees of cyclotomic algebras

Author: Gunter Malle
Journal: Represent. Theory 4 (2000), 342-369
MSC (2000): Primary 20C08, 20C40
Published electronically: August 1, 2000
MathSciNet review: 1773866
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We determine the generic degrees of cyclotomic Hecke algebras attached to exceptional finite complex reflection groups. The results are used to introduce the notion of spetsial reflection group, which in a certain sense is a generalization of the finite Weyl group. In particular, to spetsial $W$ there is attached a set of unipotent degrees which in the case of a Weyl group is just the set of degrees of unipotent characters of finite reductive groups with Weyl group $W$, and in general enjoys many of their combinatorial properties.

References [Enhancements On Off] (What's this?)

    AL D. Alvis and G. Lusztig, The representations and generic degrees of the Hecke algebra of type $H_{4}$, J. Reine Angew. Math. 336 (1982), 201–212; correction, ibid. 449 (1994), 217–218. ; BC C. T. Benson and C. W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273; corrections and additions, ibid. 202 (1975), 405–406. ;
  • Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
  • BMM M. Broué, G. Malle and J. Michel, Towards spetses II, in preparation.
  • Michel Broué, Gunter Malle, and Raphaël Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190. MR 1637497
  • Michel Broué and Jean Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 73–139 (French). MR 1429870
  • GIM M. Geck, L. Iancu and G. Malle, Weights of Markov traces and generic degrees, Indag. Mathem. (2000) (to appear). GP M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford University Press, Oxford, 2000. Lua G. Lusztig, A class of irreducible representations of a Weyl group, Indag. Mathem. 41 (1979), 323–335; II, ibid. 44 (1982), 219–226. ;
  • George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
  • George Lusztig, Appendix: Coxeter groups and unipotent representations, Astérisque 212 (1993), 191–203. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235835
  • MaU G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), 768–826.
  • Gunter Malle, Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 311–332 (French). MR 1429878
  • Gunter Malle, Spetses, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 87–96. MR 1648059
  • MaR ---, On the rationality and fake degrees of characters of cyclotomic algebras, J. Math. Sci. Univ. Tokyo 6 (1999), 647–677.
  • Giovanni Cutolo, John C. Lennox, Silvana Rinauro, Howard Smith, and James Wiegold, On infinite core-finite groups, Proc. Roy. Irish Acad. Sect. A 96 (1996), no. 2, 169–175. MR 1641198

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20C08, 20C40

Retrieve articles in all journals with MSC (2000): 20C08, 20C40

Additional Information

Gunter Malle
Affiliation: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany
MR Author ID: 225462

Received by editor(s): October 28, 1999
Received by editor(s) in revised form: June 19, 2000
Published electronically: August 1, 2000
Additional Notes: I’m grateful to J. Michel for spotting some inaccuracies in an earlier version of this paper.
I thank the Science University of Tokyo for its hospitality and the Deutsche Forschungsgemeinschaft for financial support
Article copyright: © Copyright 2000 American Mathematical Society