Advertisement

Journal of Algebraic Combinatorics

, Volume 35, Issue 2, pp 215–235 | Cite as

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

  • J. Matthew Douglass
  • Götz Pfeiffer
  • Gerhard Röhrle
Article

Abstract

In our recent paper (Douglass et al. arXiv:1101.2075 (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik–Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

Keywords

Coxeter groups Reflection arrangements Descent algebra Dihedral groups 

References

  1. 1.
    Bergeron, F., Bergeron, N., Garsia, A.M.: Idempotents for the free Lie algebra and q-enumeration. In: Invariant Theory and Tableaux, Minneapolis, MN, 1988, IMA Vol. Math. Appl., vol. 19, pp. 166–190. Springer, New York (1990) Google Scholar
  2. 2.
    Bergeron, F., Bergeron, N., Howlett, R.B., Taylor, D.E.: A decomposition of the descent algebra of a finite Coxeter group. J. Algebr. Comb. 1(1), 23–44 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brieskorn, E.: Sur les groupes de tresses [d’après V. I. Arnol’d]. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., vol. 317, pp. 21–44. Springer, Berlin (1973) CrossRefGoogle Scholar
  4. 4.
    Douglass, J.M., Pfeiffer, G., Röhrle, G.: Coxeter arrangements and Solomon’s descent algebra. arXiv:1101.2075 (2011)
  5. 5.
    Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras. London Mathematical Society Monographs. New Series, vol. 21. Clarendon Press, New York (2000) zbMATHGoogle Scholar
  6. 6.
    Hanlon, P.: The action of S n on the components of the Hodge decomposition of Hochschild homology. Mich. Math. J. 37(1), 105–124 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Howlett, R.B.: Normalizers of parabolic subgroups of reflection groups. J. Lond. Math. Soc. 21, 62–80 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Konvalinka, M., Pfeiffer, G., Röver, C.: A note on element centralizers in finite Coxeter groups. J. Group Theory (2011). doi: 10.1515/JGT.2011.074, arXiv:1005.1186 Google Scholar
  9. 9.
    Lehrer, G.I., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104(2), 410–424 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Orlik, P., Solomon, L.: Coxeter arrangements. In: Singularities, Part 2, Arcata, Calif., 1981. Proc. Sympos. Pure Math., vol. 40, pp. 269–291. Amer. Math. Soc, Providence (1983) Google Scholar
  11. 11.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin, (1992) zbMATHGoogle Scholar
  12. 12.
    Pfeiffer, G.: A quiver presentation for Solomon’s descent algebra. Adv. Math. 220(5), 1428–1465 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pfeiffer, G., Röhrle, G.: Special involutions and bulky parabolic subgroups in finite Coxeter groups. J. Aust. Math. Soc. 79(1), 141–147 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Schocker, M.: Über die höheren Lie-Darstellungen der symmetrischen Gruppen. Bayreuth. Math. Schr. 63, 103–263 (2001) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264 (1976) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • J. Matthew Douglass
    • 1
  • Götz Pfeiffer
    • 2
  • Gerhard Röhrle
    • 3
  1. 1.Department of MathematicsUniversity of North TexasDentonUSA
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  3. 3.Fakultät für MathematikRuhr-Universität BochumBochumGermany

Personalised recommendations