  In the 80's, Regev, using results of Formanek, Procesi and Razmyslov in
invariant theory and Hilbert series', determined asymptotically the codimension
sequence of mXm matrices over an algebraically closed field of characteristic
zero. Inspired by Regev's ideas, we found that the asymptotics of
$c_{n}^{G}(A)$, the G graded codimension sequence of a finite dimensional G
simple algebra A, is equal to $\alpha n^{\frac{1-\dim(A_{e})}{2}}(\dim(A)^{n}
  $ (this was conjectured by E.Aljadeff, D.Haile and M. Natapov), where \alpha
is not yet determined number. Moreover, in the case where A is the algebra of
mXm matrices with an arbitrary elementary G-grading we also manged to calculate
\alpha.
