Representations and Tensor Product Growth

Abstract

Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper, we obtain two-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 $G$ that 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.