Representations and tensor product growth

TL;DR

This paper extends product growth results from subsets to tensor products of representations in finite simple groups of Lie type, revealing new growth phenomena and covering properties for characters, and also explores similar results for compact Lie groups.

Contribution

It establishes 2-step growth results for tensor products of representations, including unbounded rank groups, and investigates character covering phenomena and growth in compact Lie groups.

Findings

Tensor products of certain characters grow significantly in size.

High-power products of characters cover all irreducible characters.

Growth results also apply to compact semisimple Lie groups.

Abstract

The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper we obtain 2-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $\chi$ a character of $G$. Let $|\chi|$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ which are constituents of $\chi$. We show that for all $\delta>0$ there exists $\epsilon>0$, independent of $G$, such that if $\chi$ is an irreducible character of $G$ satisfying $|\chi| \le |G|^{1-\delta}$, then $|\chi^2| \ge |\chi|^{1+\epsilon}$. We also obtain results for reducible characters, and establish faster growth in the case where $|\chi| \le |G|^{\delta}$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $G$ occurs as a constituent of certain products of characters. For example, we prove that if $|\chi_1| \cdots |\chi_m|$ is a high enough power of $|G|$, then every irreducible character of $G$ appears in $\chi_1\cdots\chi_m$. Finally, we obtain growth results for compact semisimple Lie groups.