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Riemann-Roch formulae for group representations

  • C. B. Thomas (a1)

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Our investigation starts from the question: is the even dimensional cohomology of the non-abelian group of order p3 and exponent p generated by Chern classes? From the computation of the complete cohomology ring in [8] one quickly sees that the essential problem is to express elements of the form cor (γk), γH2 (k, ℤ), K a subgroup of index p, in terms of Chern classes. For a more general pair of groups (KG) it is known, see [7] that the best for which one can hope is a description of some multiple of cor (γk) in this way. Our first theorem shows that, under suitable hypotheses (satisfied in particular by the example of order p3) the numerical factor may be removed. Thus

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1. Atiyah, M. F. and Hirzebruch, F., “The Riemann-Roch theorem for analytic embeddings”, Topology, 1 (1962), 151166.
2. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956).
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6. Huppert, B., Endliche Gruppen I, Die Grundlehren der Math. Wiss. 134 (Springer-Verlag, Berlin, 1968).
7. Knopfmacher, J., “Chern classes of representations of finite groups”, J. London Math. Soc., 41 (1956), 535541.
8. Lewis, O., “The integral cohomology rings of groups of order p3”, Trans. Amer. Math. Soc., 132 (1968), 501529.
9. Nakaoka, M., “Decomposition theorem for homology groups of symmetric groups”, Ann. of Math., 71 (1960), 1642.
10. Nakaoka, M., “Homology of the infinite symmetric group”, Ann. of Math., 73 (1961), 229257.
11. Thomas, C. B., “Chern classes of metacyclic p-groups”, Mathematika, 18 (1971), 196200.
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