Testing Properties of Functions on Finite Groups

We study testing properties of functions on finite groups. First we consider functions of the form f : G → 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 → Md(C), where d is a fixed constant and Md(C) is the family of d by d matrices with each element in C. For a function g : G → Md(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 for any x ∈ G.