Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by means of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non-vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codi-mensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. If c n (A),n = 1, 2,..., is the sequence of graded codimensions of A, we prove that exp (A) = lim_n-∞co ycn (A), the G-exponent of A, exists and is an integer. This result was proved by the authors and D. La Mattina in 2011 and by the second author and D. La Mattina in 2010 in the case G is abelian.