Tensor Power Asymptotics for Linearly Reductive Groups

TL;DR

This paper investigates the asymptotic behavior of the number of irreducible factors in tensor powers of a faithful representation of a linearly reductive group, revealing bounds related to the group's unipotent subgroup dimension.

Contribution

It establishes precise asymptotic bounds for the growth of irreducible factors in tensor powers of representations of linearly reductive groups, linking to the group's unipotent subgroup dimension.

Findings

Bounds are proportional to n^{-u/2} (dim V)^n

Growth rate depends on the maximal unipotent subgroup dimension

Provides asymptotic formulas for irreducible factor counts

Abstract

Given a finite-dimensional faithful representation $V$ of a linearly reductive group $G$ over a field $K=\bar K$, we consider the growth of the number of irreducible factors of $V^{\otimes n}$ when $n$ is large. We prove that there exist upper and lower bounds which are constant multiples of $n^{-u/2} (\dim V)^n$, where $u$ is the dimension of any maximal unipotent subgroup of $G$.