Skip to main content
SpringerLink
Log in

The decomposition numbers of the hecke algebra of typeF 4

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

LetW be the finite Coxeter group of typeF 4, andH r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF 4 over the ringA = ℤ[u 1/2,u -1/2] of Laurent polynomials in an indeterminateu 1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, New York, 1985

    MATH  Google Scholar 

  2. Char, B.W., Geddes, K.O., Gonnet, G.H., Monagan, M.B., Watt, S.M.: MAPLE—Reference Manual 5th edition. University of Waterloo, 1988

  3. Feit, W.: The representation theory of finite groups. North-Holland Publishing Company, 1982

  4. Fleischmann, P.: Projective simple modules of symmetric algebras and their specializations with applications to Hecke algebras. Arch. Math. 55 (1990), 247–258

    Article  MATH  MathSciNet  Google Scholar 

  5. Geck, M.: Brauer trees of Hecke algebras. Manuscript, Aachen (1990)

  6. Gyoja, A. On the existence of aW-graph for an irreducible representation of a Coxeter group. J. Algebra 86 (1984), 422–438

    Article  MATH  MathSciNet  Google Scholar 

  7. Gyoja, A., Uno, K.: On the semisimplicity of Hecke algebras. J. Math. Soc. of Japan 41(1) (1989), 75–79

    Article  MATH  MathSciNet  Google Scholar 

  8. Hiß, G.: Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nichtdefinierender Charakteristik. Habilitationsschrift, RWTH Aachen, 1989

  9. Kernighan, B.W., Ritchie, D.M.: Programmieren in C, 2. Ausgabe. Carl Hanser Verlag, München, 1990

    MATH  Google Scholar 

  10. Lusztig, G.: Cells in Affine Weyl Groups. Advanced Studies in Pure Mathematics 6 (1985), 255–287

    MathSciNet  Google Scholar 

  11. Niemeyer, A., Nickel, W., Schönert, M.: GAP—Getting started and Reference Manual, RWTH Aachen, 1988

  12. Parker, R.A.: The Computer Calculation of Modular Characters (the Meat-Axe).In: Atkinson, M.D. (ed.): Computational Group Theory, Academic Press, London, 1984

    Google Scholar 

  13. Shi, Jian-Yi: The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups. Lecture Notes in Mathematics 1179, Springer, 1980

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-5100, Aachen

    Meinolf Geck & Klaus Lux

Authors
  1. Meinolf Geck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Klaus Lux
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Geck, M., Lux, K. The decomposition numbers of the hecke algebra of typeF 4 . Manuscripta Math 70, 285–306 (1991). https://doi.org/10.1007/BF02568379

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02568379

Keywords

Search

Navigation