Growth Problems for Representations of Finite Groups

TL;DR

This paper derives a general asymptotic formula for the growth of indecomposable summands in tensor powers of finite group representations, providing exact results in characteristic zero and computational tools.

Contribution

It introduces a new asymptotic formula for representation growth and offers exact results in characteristic zero, along with computational methods and examples.

Findings

Asymptotic growth formula for indecomposable summands

Exact growth rate formula in characteristic zero

Computational code and examples included

Abstract

We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain additional results, including an exact formula for the growth rate. We compute various examples and also provide code used to compute our formulas.