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.