Representations and Tensor Product Growth

Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets finite simple groups $G$ Lie type bounded rank. In this paper, we obtain two-step results representations such (including those unbounded rank), where products are replaced by tensor representations. Let be a group and $\chi $ character $G$. $|\chi |$ denote the sum squares degrees all (distinct) irreducible characters that constituents $. We show $\delta>0$, there exists $\epsilon>0$, independent $G$, if is an satisfying | \le |G|^{1-\delta }$, then ^2| \ge |\chi |^{1+\epsilon }$. also reducible establish faster in case |G|^{\delta another direction, explore covering phenomena, namely situations every occurs as constituent certain characters. For example, prove _1| \cdots _m|$ high enough power $|G|$, appears _1\cdots \chi _m$. Finally, compact semisimple groups.