Approximate Representations and Approximate Homomorphisms

TL;DR

This paper investigates approximate representations of finite groups, establishing bounds on their quality based on the ratio of the representation dimension to the smallest nontrivial representation, with implications for approximate homomorphisms and quasirandom groups.

Contribution

It provides new bounds on approximate representations and homomorphisms of finite groups, linking quasirandomness to embedding limitations and demonstrating the tightness of these bounds.

Findings

Bounds relate approximation quality to the ratio d/d_min

Approximate homomorphisms are limited when target groups have smaller representations

Bounds are shown to be tight through analysis of genuine representations

Abstract

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensions, or in a less-quasirandom group, without significant distortion of G's multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations.