Morita Equivalence, GLn(Fq)-Modules, and the Steenrod Algebra
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.
KeywordsUnstable Module Full Subcategory Algebraic Topology Privileged Role Mathematical Science Research Institute
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- [AF]F.W. Anderson and K.R. Fuller, Rings and categories of modules, Springer Graduate Texts in Math. 13, 1974.Google Scholar
- [FS]V. Franjou and L. Schwartz, Reduced unstable A-modules and the modular representation theory of the symmetric groups, Preprint (1989).Google Scholar
- [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
- [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
- [LS2]J. Lannes and L. Schwartz, Polynomial Functors and unstable A-modules, in preparation.Google Scholar
- [Mac]I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monographs, Oxford University Press, New York, 1979.Google Scholar
- [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