Approximate Representations, Approximate Homomorphisms, and Low-Dimensional Embeddings of Groups

Approximate algebraic structures play a defining role in additive number theory and have found remarkable applications to questions in theoretical computer science, including in pseudorandomness and probabilistically checkable proofs. Here we study approximate representations of finite groups: functions $\psi : G \to \textsf{U}_d$ such that $\Pr[\psi(xy) = \psi(x) \,\psi(y)]$ is large or, more generally, such that the expected $\ell_2$ norm squared $\mathbb{E}_{x,y} \left\| \psi(xy) - \psi(x) \,\psi(y) \right\|_2^2$ is small, where $x, y$ are uniformly random elements of the group $G$ and $\textsf{U}_d$ denotes the group of unitary operators on $\mathbb{C}^d$. 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 $f:G \to H$ can be an approximate homomorphism where $H$ is another finite group. We show that if $H$'s representations are significantly smaller...