2021/04/23 by Larsen, Michael, Shalev, Aner, Tiep, Pham Huu
#20C33 #20D06 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2104.11716
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 χ a character of G. Let |χ| denote the sum of the squares of the degrees of all (distinct) irreducible characters of G which are constituents of χ. We show that for all δ>0 there exists ε>0, independent of G, such that if χ is an irreducible character of G satisfying |χ| ≤ |G|1-δ, then |χ2| ≥ |χ|1+ε. We also obtain results for reducible characters, and establish faster growth in the case where |χ| ≤ |G|δ. 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 |χ1| ⋯ |χm| is a high enough power of |G|, then every irreducible character of G appears in χ1⋯χm. Finally, we obtain growth results for compact semisimple Lie groups.