Representations and Tensor Product Growth

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