Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T15:51:00.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings of Invariants and p-Sylow Subgroups

Published online by Cambridge University Press:  20 November 2018

H. E. A. Campbell ,
I. Hughes  and
R. D. Pollack
H. E. A. Campbell
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6
I. Hughes
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6
R. D. Pollack
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6
Rights & Permissions [Opens in a new window]

Abstract

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.

Let V be a vector space of dimension n over a field k of characteristic p. Let G ⊆ Gl(V) be a finite group with p-Sylow subgroup P. G and P act on the symmetric algebra R of V. Denote the respective rings of invariants by RG and Rp. We show that if Rp is Cohen-Macaulay (CM) so also is RG, generalizing a result of M. Hochster and J. A. Eagon. If P is normal in G and G is generated by P and pseudo-reflections, we show that if RG is CM so also is Rp. However, in general, RG may even be polynomial with Rp not CM. Finally, we give a procedure for determining a set of generators for RG given a set of generators for Rp.

Keywords

Invariant theory13F20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 1 , 01 March 1991 , pp. 42 - 47
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. H. Campbell, E. A., Hughes, I., Pollack, R. D., On the vector invariants of the symmetric groups preprint.Google Scholar
2. Fossum, R. M., Griffith, P. A., Complete local factorial rings which are not Cohen-Mac aulay in characteristic p, Ann. scient. Èc. Norm. Sup. 4e série 8(1975) 189–200.Google Scholar
3. Hochster, M., The invariant theory of commutative rings, Contemp. Math. 43(1985) 161–179.Google Scholar
4. Hochster, M., Eagon, J. A., Cohen-Mac aulay rings, invariant theory, and the generic perfection ofdeterminantal loci, Amer. J. Math. 93(1971) 1020–1058.Google Scholar
5. Lam, T. Y., Serre 's conjecture, Lecture Notes in Math. 635 Springer-Verlag New York.Google Scholar
6. Mùi, H., Modular invariant theory and the cohomology algebras of symmetric groups, J. Fac. Sci., Univ. Tokyo, Sec. IA, 22 (1975) 319–369.Google Scholar
7. Nakajima, H., Invariants of finite abelian groups generated by transvections, Tokyo Journal of Math. (2)3 (1980) 201–214.Google Scholar
8. Nakajima, H., Regular rings of invariants ofunipotent groups, Journal of Algebra 85 (1986) 253–286.Google Scholar
9. Serre, J. P., Algèbre Locale Multiplicités, Lecture Notes in Math. 11 Springer-Verlag, New York 1975 .Google Scholar
10. Stanley, R. P., Invariants of finite groups and their applications to combinatorics, Bull. A.M.S. (3)1 May 1979 475–511.Google Scholar
11. Weyl, H., Classical Groups, Princeton University Press, Princeton, New Jersey, U.S.A. 1939.Google Scholar
12. Wilkerson, C., A primer on the Dickson invariants, Contemp. Math. 19(1983) 421–434.Google Scholar
13. Zariski, O., Samuel, P., Commutative Algebra, Vol. II van Nostrand Princeton, New Jersey, U.S.A. 1960.Google Scholar