On the Codimension Sequence of G-Simple Algebras

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.