Let $F$ be an algebraically closed field of characteristic $0$ , and let $A$ be a $G$ -graded algebra over $F$ for some finite abelian group $G$ . Through $G$ being regarded as a group of automorphisms of $A$ , the duality between graded identities and $G$ -identities of $A$ is exploited. In this framework, the space of multilinear $G$ -polynomials is introduced, and the asymptotic behavior of the sequence of $G$ -codimensions of $A$ is studied.

Two characterizations are given of the ideal of $G$ -graded identities of such algebra in the case in which the sequence of $G$ -codimensions is polynomially bounded. While the first gives a list of $G$ -identities satisfied by $A$ , the second is expressed in the language of the representation theory of the wreath product $G \wr S_n$ , where $S_n$ is the symmetric group of degree $n$ .

As a consequence, it is proved that the sequence of $G$ -codimensions of an algebra satisfying a polynomial identity either is polynomially bounded or grows exponentially.