Regarding a Representation-Theoretic Conjecture of Wigderson

TL;DR

This paper disproves Wigderson's conjecture by demonstrating a family of irreducible representations where the average over random elements consistently has operator norm 1, with probability approaching 1 as the group size increases.

Contribution

It provides a counterexample to a conjecture in representation theory, showing the existence of specific irreducible representations with particular averaging properties.

Findings

Disproves Wigderson's conjecture.

Constructs a family of irreducible representations with operator norm 1 on average.

Shows probability approaches 1 as group size increases.

Abstract

We show that there exists a family of irreducible representations R_i (of finite groups G_i) such that, for any constant t, the average of R_i over t uniformly random elements g_1, ..., g_t of G_i has operator norm 1 with probability approaching 1 as i limits to infinity. This settles a conjecture of Wigderson in the negative.