# The non-projective part of the tensor powers of a module

## TL;DR

This paper studies a mathematical invariant related to tensor powers of modules over finite groups, exploring its properties using representation theory and Banach algebra techniques.

## Contribution

It introduces and analyzes the invariant gamma_G(M), connecting representation theory with Banach algebra methods to understand tensor power behaviors.

## Key findings

- Gamma_G(M) is a well-defined invariant with specific properties.
- The invariant relates to the radius of convergence of a generating function.
- Connections between representation theory and Banach algebra are established.

## Abstract

Let $M$ be a finite dimensional modular representation of a finite group $G$. We consider the generating function for the non-projective part of the tensor powers of $M$, and we write $\gamma_G(M)$ for the reciprocal of the radius of convergence of this power series. We investigate the properties of the invariant $\gamma_G(M)$, using tools from representation theory, and from the theory of commutative Banach algebras.