Abstract

In modular representation theory of finite groups, one of the big
mysteries is the structure of tensor products of modules, with the
diagonal group action. In particular, given a module $M$, we can look
at the tensor powers of $M$ and ask about the asymptotics of how
they decompose. For this purpose, we introduce an new invariant
$\gamma(M)$ and investigate some of its properties. Namely, we
write $c_n(M)$ for the dimension of the non-projective part of
$M^{$, and $\gamma_G(M)$ for $\frac{1}{r}$”, where $r$ is the
radius of convergence of the generating function $\sum z}n c_n(M)$.
The properties of the invariant $\gamma(M)$ are controlled by a
certain infinite dimensional commutative Banach algebra associated
to $kG$. This is joint work with Peter Symonds. We end with a number
of conjectures and directions for further research.