Representations and tensor product growth

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.

18 pages