Extended integrity bases of finite groups

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Abstract
An extended integrity basis (EIB) of a polynomial algebra in a set of variables on which a finite group operates includes the ordinary integrity basis of invariants and linear integrity bases of covariants. The latter are defined as sets of covariants of a given type such that any other covariant of this type is expressible as a linear combination of basic ones with invariants as coefficients of this combination. A constructive method of derivation, based on successive Clebsch-Gordan reaction and elimination of redundant covariants, is described, and the 'extended Noether's theorem', which states that the EIB of a finite group in a finite set of variables is finite, is proved with its use. It is shown that EIBs in irreducible sets of variables are fundamental for a given group because overall homogeneous EIBs in any set of variables can be constructed with their use for this group. A relationship between this method and theory based on a consideration of Molien series is established. It is shown that the division of invariants into denominator and numerator invariants enables one to construct general invariant functions as well as functional covariants.