Group actions and asymptotic behavior of graded polynomial identities

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 GintegralS(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.