On rings of invariants of non-modular Abelian groups

Abstract

We study the ring of invariant Laurent polynomials associated to the action of a finite diagonal group G on the symmetric algebra of a vector space over a field F. Here the characteristic p of the field F necessarily does not divide the order q = |G| of the group, so G is said to be non-modular. For certain representations of such groups, we can characterise generators of the ring of invariant polynomials in the original symmetric algebra, extending results of Campbell, Hughes, Pappalardi and Selick. In particular we obtain a recursive formula for the number of minimal generators for these rings of invariants.