Graded algebras with polynomial growth of their codimensions

Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G. We study combinatorial and asymptotic properties of the G-graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G-graded algebra in the variety generated by A. We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row. We relate, moreover, the polynomial growth to the colengths. Finally we describe in detail the algebras whose graded codimensions are of linear growth.