Group representations that resist random sampling

TL;DR

This paper constructs families of groups with irreducible representations that maintain high operator norm even after averaging over many random elements, answering a conjecture about the resistance of certain representations to random sampling.

Contribution

It demonstrates the existence of groups with representations that resist norm reduction under random sampling, settling a conjecture by Wigderson.

Findings

Existence of groups with representations requiring logarithmic double-logarithmic samples to reduce norm

Operator norm remains close to 1 with high probability after averaging over few random elements

Resistant representations challenge assumptions about random sampling effectiveness

Abstract

We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $\rho_n$ such that, for any constant $t$, the average of $\rho_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has operator norm $1$ with probability approaching 1 as $n \rightarrow \infty$. More quantitatively, we show that there exist families of finite groups for which $\Omega(\log \log |G|)$ random elements are required to bound the norm of a typical representation below $1$. This settles a conjecture of A. Wigderson.