Asymptotic Growth of Trivial Summands in Tensor Powers

TL;DR

The paper analyzes the exponential growth rate of trivial summands in tensor powers $V^{\otimes n}$ of a finite-dimensional $G$-representation, defining $d(G,V)$ as the limsup of the nth root of their multiplicities. In characteristic zero, it establishes that $d(G,V)=\dim V$ precisely when the determinant map on $\rho(G)$ takes finitely many values, with SL$(V)$ achieving the maximal rate; this result extends to arbitrary $G$ via determinant considerations. In positive characteristic, a parallel criterion emerges: $d(G,V)=\dim V$ iff the determinant values are finite and the unipotent subset of the Zariski closure is trivial; otherwise the growth rate is strictly smaller. The analysis combines weight theory, Weyl dimension bounds, and concentration inequalities, with a modular representation analysis of the cyclic group of order $p$ as a key step for the general case.

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.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Definition
• Lemma 4.1
• proof
• Lemma 4.2