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

Abstract

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 ${\alpha }_1{n}^{\beta }{l}^n\le c_n^G(W)\le {\alpha }_2{n}^{\beta }{l}^n$. Furthermore, if W has a unit then the asymptotic behavior of $c_n^G(W)$ is $\alpha n^{\beta }{l}^n$ where $\alpha \in \mathbb{R},\beta \in \frac{1}{2}\mathbb{Z},l\in \mathbb{N}$.