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On the Chern classes of the regular representations of some finite groups

Published online by Cambridge University Press:  20 January 2009

Benjamin M. Mann  and
R. James Milgram
Benjamin M. Mann
Affiliation:
Bowdoin College, Stanford University
R. James Milgram
Affiliation:
Bowdoin College, Stanford University
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In studying the cohomology of the symmetric groups and its applications in topology one is led to certain questions concerning the representation rings of special subgroups of . In this note we calculate the Chern classes of the regular representation of (Z/p)n where p is a fixed odd prime in terms of certain modular invariants first described by L. E. Dickson in 1911. In a later paper [9] we apply these results to study the odd primary torsion in the PL cobordism ring. Some indications of this application are given in Sections 10–12 where we apply the result above to obtain information about the cohomology of . After circulation of this note in preprint form we learned that H. Mui [10], has also proved Theorem 6.2.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 25 , Issue 3 , October 1982 , pp. 259 - 268
Copyright
Copyright © Edinburgh Mathematical Society 1982

References

REFERENCES

1.Dickson, L. E., A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75–98.CrossRefGoogle Scholar
2.Green., J. A., The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447.CrossRefGoogle Scholar
3.Lang, S., Algebra (Addison Wesley, 1965).Google Scholar
4.Mann., B. M., The cohomology of the symmetric groups, Trans. Amer. Math. Soc. 242 (1978), 157–184.CrossRefGoogle Scholar
5.Moore., E. H., A two-fold generalization of Fermat's theorem, Bull. Amer. Math. Soc. 2nd series 2 (1896), 189–199.CrossRefGoogle Scholar
6.Quillen., D., The Adams conjecture, Topology 10 (1971), 67–80.CrossRefGoogle Scholar
7.Quillen., D., On the cohomology and K-theory of the general linear group over a finite field, Ann. Math. 96 (1972), 552–586.CrossRefGoogle Scholar
8.Steenrod., N. E. and Epstein, D. B. A., Cohomology Operations (Ann. of Math. Studies 50, Princeton U. Press, 1962).Google Scholar
9.Mann, B. M. and Milgram, R. J.. On the action of the Steenrod algebra, A(p) on H*(MSPL, Z/p) at odd primes (preprint, Stanford University, 1976).Google Scholar
10.Mui, H., Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo 22 (1975), 319–369.Google Scholar