On the codimension sequence of G-simple algebras

Let F be an algebraically closed field of characteristic zero and let G   be a finite group. In this paper we will show that the asymptotics of cnG(A), the G-graded codimension sequence of a finite dimensional G-simple F-algebra A  , has the form αn1−dimF⁡Ae2(dimF⁡A)n (as conjectured by E. Aljadeff, D. Haile and M. Natapov), where α   is some positive real number and AeAe denotes the identity component of A. In the special case where A   is the algebra of m×mm×m matrices with an arbitrary elementary G-grading we succeeded in calculating the constant α explicitly.