Tensor Quasi-Random Groups

TL;DR

This paper extends Gowers' characterization of quasi-random groups to tensor quasi-random groups, providing a dual perspective involving tensor products of representations instead of subset multiplication.

Contribution

It introduces a dual characterization of tensor quasi-random groups, broadening the understanding of group properties through representation theory.

Findings

Characterization of tensor quasi-random groups via tensor products

Connection between irreducible representations and tensor quasi-randomness

Extension of Gowers' results to a new algebraic framework

Abstract

Gowers has elegantly characterized the finite groups $G$ in which $A_1A_2A_3 = G$ for any positive density subsets $A_1,A_2,A_3$. This property, quasi-randomness, holds if and only if G does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of tensor quasi-random groups in which multiplication of subsets is replaced by tensor product of representations.