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Modular representations of abelian groups with regular rings of invariants

Published online by Cambridge University Press:  22 January 2016

Haruhisa Nakajima
Haruhisa Nakajima*
Affiliation:
Keio University, Present Address: Department of Mathematics Tokyo Metropolitan University Fukasawa, Setagaya-kn, Tokyo 158, Japan
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Let k be a field of characteristic p and G a finite subgroup of GL(V) where V is a finite dimensional vector space over k. Then G acts naturally on the symmetric algebra k[V] of V. We denote by k[V]G the subring of k[V] consisting of all invariant polynomials under this action of G. The following theorem is well known.

Theorem 1.1 (Chevalley-Serre, cf. [1, 2, 3]). Assume that p = 0 or (|G|, p) = 1. Then k[V]G is a polynomial ring if and only if G is generated by pseudo-reflections in GL(V).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 86 , June 1982 , pp. 229 - 248
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Bourbaki, N., Groupes et algebres de Lie, Ch. 4, 5 et 6, Hermann, Paris, 1968.Google Scholar
[ 2 ] Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math., 77 (1955), 778–782.Google Scholar
[ 3 ] Serre, J.-P., Groupes finis d’automorphismes d’anneaux locaux réguliers, Colloq. d’Alg. E.N.S., (1967).Google Scholar