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 abeliangroup G. Through G being regarded as a group of automorphismsof A, the duality between graded identities and G-identitiesof A is exploited. In this framework, the space of multilinearG-polynomials is introduced, and the asymptotic behavior ofthe sequence of G-codimensions of A is studied. Two characterizations are given of the ideal of G-graded identitiesof such algebra in the case in which the sequence of G-codimensionsis polynomially bounded. While the first gives a list of G-identitiessatisfied by A, the second is expressed in the language of therepresentation theory of the wreath product G S_n, where S_nis the symmetric group of degree n. As a consequence, it is proved that the sequence of G-codimensionsof an algebra satisfying a polynomial identity either is polynomiallybounded or grows exponentially.