Testing Properties of Functions on Finite Groups

TL;DR

This paper investigates the testability of various properties of functions on finite groups, including conjugate invariance, homomorphism, and matrix isomorphism, using a constant number of queries based on representation theory and harmonic analysis.

Contribution

It introduces new property testing results for functions on finite groups, extending to matrix-valued functions and utilizing harmonic analysis techniques.

Findings

Conjugate invariance, homomorphism, and character proportionality are testable with constant queries.

Matrix-valued functions' unitary isomorphism can be tested efficiently.

The methods rely on representation theory and harmonic analysis on finite groups.

Abstract

We study testing properties of functions on finite groups. First we consider functions of the form $f:G \to \mathbb{C}$, where $G$ is a finite group. We show that conjugate invariance, homomorphism, and the property of being proportional to an irreducible character is testable with a constant number of queries to $f$, where a character is a crucial notion in representation theory. Our proof relies on representation theory and harmonic analysis on finite groups. Next we consider functions of the form $f: G \to M_d(\mathbb{C})$, where $d$ is a fixed constant and $M_d(\mathbb{C})$ is the family of $d$ by $d$ matrices with each element in $\mathbb{C}$. For a function $g:G \to M_d(\mathbb{C})$, we show that the unitary isomorphism to $g$ is testable with a constant number of queries to $f$, where we say that $f$ and $g$ are unitary isomorphic if there exists a unitary matrix $U$ such that $f(x) = Ug(x)U^{-1}$ for any $x \in G$.