Let $G$ be a finite group and $k$ be a field of characteristic $p$. If $M$ is a finite-dimensional $kG$-module, we write $c_n(M)$ for the dimension of the non-projective part of $M^{\otimes n}$, and $\gamma_G(M) for $\frac{1}{r}$, where $r$ is the radius of convergence of the generating function $\sum z^n c_n(M)$. We investigate the properties of the invariant $\gamma_G$, and its relationship with a certain commutative Banach algebra associated to $kG$. (Joint work with Peter Symonds.)