Quantum algorithms ( CO 781 / CS 867 / QIC 823 , Winter 2013 )

To understand nonabelian Fourier analysis, we first need to introduce some notions from group representation theory. For further information on this subject, a good basic reference is the book Linear Representations of Finite Groups by Serre. A linear representation (or simply representation) of a group G over the vector space Cn is a homomorphism σ : G → GL(Cn), i.e., a map from group elements to nonsingular n × n complex matrices satisfying σ(x)σ(y) = σ(xy) for all x, y ∈ G. Clearly, σ(1) = 1 and σ(x−1) = σ(x)−1. We call Cn the representation space of σ, where n is called its dimension (or degree), denoted dσ. Two representations σ and σ′ with representation spaces Cn are called isomorphic (denoted σ ∼ σ′) if there is an invertible linear transformation M ∈ Cn×n such that Mσ(x) = σ′(x)M for all x ∈ G. Otherwise they are called non-isomorphic (denoted σ 6∼ σ′). In particular, representations of different dimensions are non-isomorphic. Every representation of a finite group is isomorphic to a unitary representation, i.e., one for which σ(x)−1 = σ(x)† for all x ∈ G. Thus we can restrict our attention to unitary representations without loss of generality. The simplest representations are those of dimension one, such that σ(x) ∈ C with |σ(x)| = 1 for all x ∈ G. Every group has a one-dimensional representation called the trivial representation, defined by σ(x) = 1 for all x ∈ G.