Asymptotic Growth of Trivial Summands in Tensor Powers

TL;DR

This paper investigates the asymptotic behavior of trivial summands in tensor powers of finite-dimensional group representations, providing conditions for their growth in different characteristic fields.

Contribution

It establishes necessary and sufficient conditions for the existence of subsequences of tensor powers with abundant trivial summands in characteristic zero and positive characteristic fields.

Findings

Characterizes when trivial summands grow in tensor powers

Provides criteria for subsequences with many trivial summands

Applies to representations over algebraically closed fields

Abstract

Given a finite-dimensional representation $V$ over an algebraically closed field of an abstract group $G$, we consider the number of the trivial summand counted with multiplicity in the direct sum decomposition of $V^{\otimes n}$. We give necessary and sufficient conditions when the field is of characteristic $0$ and when the field is of characteristic $p$ so that $(V^{\otimes n})_n$ has a subsequence $(V^{\otimes n_k})_k$ such that $V^{\otimes n_k}$ contains enough trivial summands when $k$ is sufficiently large.