Morita Equivalence, GLn(Fq)-Modules, and the Steenrod Algebra

  • Nicholas J. Kuhn
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 27)


The purpose of this announcement is to describe, on one hand, a new approach to the modular representation theory of the semigroup of n × n matrices M n (F q ) and its group of units GL n (F q ), and, on the other, a new approach to studying the category of unstable modules over the Steenrod “q th-power” operations. Perhaps most remarkable is that these two approaches are the same via a generalization of classical Morita equivalence.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AF]
    F.W. Anderson and K.R. Fuller, Rings and categories of modules, Springer Graduate Texts in Math. 13, 1974.Google Scholar
  2. [AGM]
    J.F. Adams, J.H. Gunawardena and H.R. Miller, The Segal conjecture for elementary abelian p-groups, Topology 24(1985), 435–460.CrossRefMATHMathSciNetGoogle Scholar
  3. [C]
    C. Carlsson, G.B. Segal’s Burnside ring conjecture for(ℤ2)k, Topology 22(1983), 83–103.CrossRefMATHMathSciNetGoogle Scholar
  4. [FS]
    V. Franjou and L. Schwartz, Reduced unstable A-modules and the modular representation theory of the symmetric groups, Preprint (1989).Google Scholar
  5. [HLS]
    H.W. Henn, J. Lannes and L. Schwartz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Ann Scient. Ecole Norm. Sup. 23(1990), 593–624.Google Scholar
  6. [K]
    N.J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra: I, II, III., Part I is a preprint. Parts II and III are in preparation.Google Scholar
  7. [LS]
    J. Lannes and L. Schwartz, Sur la structure des A-modules instables injectifs, Topology 28(1989), 153–169.CrossRefMATHMathSciNetGoogle Scholar
  8. [LS2]
    J. Lannes and L. Schwartz, Polynomial Functors and unstable A-modules, in preparation.Google Scholar
  9. [LZ]
    J. Lannes and S. Zarati, Foncteurs dérivés de la déstabilisation, Math. Zeit. 194(1987), 25–59.CrossRefMATHMathSciNetGoogle Scholar
  10. [Mac]
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monographs, Oxford University Press, New York, 1979.Google Scholar
  11. [Ml]
    H.R. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. Math. 120(1984), 39–87.CrossRefMATHGoogle Scholar
  12. [Mn]
    J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1958), 150–171.CrossRefMATHMathSciNetGoogle Scholar
  13. [W]
    R.M. Wood, SplittingΣ(CP∞ ×…× CP∞) and the action of Steenrod squares S q i on the polynomial ringF2[x1,..., xn],, in “Algebraic Topology”—Barcelona 1986, Springer L. N. Math. 1298(1987), 237–255.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Nicholas J. Kuhn
    • 1
  1. 1.Department Of MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations