The Asymptotic Behavior of the Codimension Sequence of Affine G - Graded Algebras

Let W be an affine PI algebra over a field of characteristic zero graded by a finite group G. We show that there exist $\alpha_{1},\alpha_{2}\in\mathbb{R}, \beta\in\frac{1}{2}\mathbb{Z}$, and $l\in\mathbb{N}$ such that $\alpha_{1}n^{\beta}l^{n}\leq c_{n}^{G}(W)\leq\alpha_{2}n^{\beta}l^{n}$. Furthermore, if W has a unit then the asymptotic behavior of $c_{n}^{G}(W)$ is $\alpha n^{\beta}l^{n}$ where $\alpha\in\mathbb{R}, \beta\in\frac{1}{2}\mathbb{Z}, l\in\mathbb{N}$.