# Covering and growth for group subsets and representations

### Aner Shalev

Hebrew University of Jerusalem, Israel

This book chapter is published *open access.*

## Abstract

Deep results on products of subsets of finite groups, and of finite simple groups in particular, were obtained this century. Gowers’ theory of quasi-random groups, further developed and applied by Nikolov and Pyber, focuses on covering results, while the theory of approximate subgroups and the product theorem, developed by Helfgott, Hrushovski, Breuillard, Green and Tao, and Pyber and Szabó, focus on growth results.

In recent joint works with Larsen and Tiep, following works with Liebeck and Tiep, we explore analogous problems in representation theory. We replace subsets of a group by its characters, and subset products by products of characters. We also study covering and growth for normal subsets of finite simple groups and derive various applications. In particular, we prove that every element of a sufficiently large finite simple transitive permutation group is a product of two derangements.

The product theorem establishes 3-step growth of the form $|A^3|\ge |A|^{1+\varepsilon}$ for (certain) subsets $A$ of finite simple groups of Lie type of bounded rank. Surprisingly, stronger results hold for characters. We obtain 2-step growth for characters of finite simple groups of Lie type, including those of unbounded rank. For a character $\chi$ of $G$, we set $|\chi|=\sum_i\chi_i(1)^2$, where $\chi_i$ are the (distinct) irreducible constituents of $\chi$. For a finite simple group $G$ of Lie type, we show that for every $\delta>0$ there exists $\varepsilon>0$ such that if $\chi$ is an irreducible character of $G$ satisfying $|\chi|\le |G|^{1-\delta}$, then $|\chi^2| \ge |\chi|^{1+\varepsilon}$. In addition, we obtain results for reducible characters and establish faster growth of the form $|\chi^2| \ge |\chi|^{2-\varepsilon}$ if $|\chi| \le |G|^{\delta}$.

Following a recent work of Sellke, we also study covering phenomena in representation theory, proving that if $|\chi_1|\cdots |\chi_m|$ is a sufficiently large power of $|G|$, then every irreducible character of $G$ is a constituent of $\chi_1\cdots\chi_m$. Finally, we obtain related results for characters of compact semisimple Lie groups.