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Vector Invariants of Symmetric Groups

Published online by Cambridge University Press:  20 November 2018

H. E. A. Campbell
Affiliation:
Mathematics and Statistics Department Queen's University Kingston, Ontario K7L 3N6
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Abstract

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Let M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Campbell, H. E. A., Hughes, I., Pollack, R. D., Rings of invariants and p-Sylow subgroups, submitted to Canad. Math. Bull.Google Scholar
2. Stanley, R. P., Invariants of finite groups and their applications to combinatorics, Bull. A. M. S. 1 (3) (1979), 475511.Google Scholar
3. Weyl, H., Classical Groups, Princeton University Press, Princeton.Google Scholar