Inverse and stability theorems for approximate representations of finite groups

Abstract

The $U^2$ norm gives a useful measure of quasirandomness for real- or complex-valued functions defined on finite (or, more generally, locally compact) groups. A simple Fourier-analytic argument yields an inverse theorem, which shows that a bounded function with a large $U^2$ norm defined on a finite Abelian group must correlate significantly with a character. In this paper we generalize this statement to functions that are defined on arbitrary finite groups and that take values in M$_n(\mathbb C)$. The conclusion now is that the function correlates with a representation -- though with the twist that the dimension of the representation is shown to be within a constant of $n$ rather than being exactly equal to $n$. There are easy examples that show that this weakening of the obvious conclusion is necessary. The proof is much less straightforward than it is in the case of scalar functions on Abelian groups. As an easy corollary, we prove a stability theorem for near representations. It states that if $G$ is a finite group and $f:G\to$M$_n(\mathbb C)$ is a function that is close to a representation in the sense that $f(xy)-f(x)f(y)$ has a small Hilbert-Schmidt norm (also known as the Frobenius norm) for every $x,y\in G$, then there must be a representation $ρ$ such that $f(x)-ρ(x)$ has small Hilbert-Schmidt norm for every $x$. Again, the dimension of $ρ$ need not be exactly $n$, but it must be close to $n$. We also obtain stability theorems for other Schatten $p$-norms. A stability theorem of this kind was obtained for the operator norm by Grove, Karcher and Ruh in 1974 and in a more general form by Kazhdan in 1982. (For the operator norm, the dimension of the approximating representation is exactly $n$.)