Advertisement

Around Solomon’s Descent Algebras

  • C. Bonnafé
  • G. PfeifferEmail author
Article

Abstract

We study different problems related to the Solomon’s descent algebra Σ(W) of a finite Coxeter group (W,S): positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if W is irreducible and if the longest element is central, then the Loewy length of Σ(W) is equal to \(\displaystyle{\left\lceil\frac{|S|}{2}\right\rceil}\).

Keywords

Solomon’s descent algebras Finite Coxeter group Loewy length 

Mathematics Subject Classifications (2000)

Primary 20F55 Secondary 05E99 

References

  1. 1.
    Atkinson, M.D.: Solomon’s descent algebra revisited. Bull. London Math. Soc. 24, 545–551 (1992)zbMATHCrossRefMathSciNetGoogle 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. Algebraic Combin. 1, 23–44 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bergeron, F., Garsia, A., Reutenauer, C.: Homomorphisms between Solomon’s descent algebras. J. Algebra 150, 503–519 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blessenohl, D., Hohlweg, C., Schocker, M.: A symmetry of the descent algebra of a finite Coxeter group. Adv. Math. 193, 416–437 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bonnafé, C.: Representation theory of Mantaci-Reutenauer algebras. Algebr. Represent. Theor. (2008). doi:10.1007/s10468-007-9074-1
  6. 6.
    Bonnafé, C., Hohlweg, C.: Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups. Ann. Inst. Fourier 56, 131–181 (2006). With an appendix of Baumann, P., Hohlweg, C., Comparison with Specht’s constructionzbMATHGoogle Scholar
  7. 7.
    Bourbaki, N.: Groupes et algèbres de Lie, Chapitres IV, V, VI. Hermann, Paris (1968)Google Scholar
  8. 8.
    Geck, M., Hiss, G., Lübeck, F., Malle, G., Pfeiffer, G.: CHEVIE—A system for computing and processing generic character tables. Appl. Algebra Engrg. Comm. Comput. 7, 175–210 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Geck, M., Pfeiffer, G.: Characters of finite Coxeter groups and Iwahori–Hecke algebras. London Mathematical Society Monographs, New Series, vol. 21. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  10. 10.
    Lusztig, G.: Coxeter orbits and eigenspaces of Frobenius. Invent. Math. 38, 101–159 (1976/77)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Mantaci, R., Reutenauer, C.: A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products. Comm. Algebra 23, 27–56 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41, 255–264 (1976)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Labo. de Math. de Besançon (CNRS: UMR 6623)Université de Franche-ComtéBesançon CedexFrance
  2. 2.Department of MathematicsNational University of IrelandGalwayIreland

Personalised recommendations