Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T05:16:17.544Z Has data issue: false hasContentIssue false
  • English
  • Français

Operations in Grothendieck Rings and the Symmetric Group

Published online by Cambridge University Press:  20 November 2018

John Burroughs
John Burroughs*
Affiliation:
University of Michigan, Ann Arbor, Michigan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 543 - 550
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Atiyah, M. F., Power operations in k-theory, Quart. J. Math. Oxford Ser. 17 (1966), 165–93.Google Scholar
2. Atiyah, M. F. and Tall, D. O., Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–97.Google Scholar
3. Berthelot, P., Grothendieck, A., and Illusie, L., Théorie des Intersections et Théorème de Riemann-Roch (SGA6), Lecture Notes in Math. 225 (Springer, Berlin, 1971).Google Scholar
4. Boerner, H., Representations of groups (North-Holland Publishing Co., Amsterdam, 1963).Google Scholar
5. Cartier, P., Groupes formels associés aux anneaux de Witt généralisés, C. R. Acad. Sci. Paris, Sér. A-B 265 (1967), 49–42.Google Scholar
6. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, (Interscience Publ., New York, 1962).Google Scholar
7. Heller, A. and Reiner, I., Grothendieck groups of orders in semi-simple algebras, Trans. Amer. Math. Soc. 112 (1964), 344–55.Google Scholar
8. Kerber, A., Representations of permutation groups. I, Lecture Notes in Mathematics 240 (Springer, Berlin, 1971).Google Scholar
9. Robinson, G. de B., Representation theory of the symmetric group (University of Toronto Press, Toronto 1961).Google Scholar
10. Swan, R., A splitting principle in algebraic K-theory, Representation theory of finite groups and related topics, Proc. Sym. in Pure Math., XXI (1971), 155–160.Google Scholar