Representation theory is Fourier analysis

At the Mathematics Department at Royal Holloway we’ve started to read Persi Diaconis’ lecture notes Group Representations in Probability and Statistics. Chapter 2 of his book presents an interesting take on representation theory, in which the essential object is a matrix-valued Fourier transform. I found it helpful to think about how this definition relates to (1) Fourier coefficients of complex functions and (2) group algebras and algebra homomorphisms.

Here is Diaconis’ definition. Let G be a finite group and let P : G \rightarrow \mathbf{C} be a function. (In practice, P is often a probability measure on G, but we don’t need this for the definition to make sense.) Let \rho : G \rightarrow \mathrm{GL}_n(\mathbf{C}) be representation of G; that is, \rho is a group homomorphism from G into the group of invertible n \times n matrices. The Fourier transform of P at the representation \rho is

\hat{P}(\rho) = \sum_{s \in G} P(s) \rho(s).

So the Fourier transform is a function from representations to matrices.

(1) All this works fine if we take G = \mathbf{R}, provided we replace the sum with an integral and require the functions P to be periodic, say with period 1. (So really we are looking at representations of the unit circle.) For each n \in \mathbf{N} we may define a representation \rho_n : \mathbf{R} \rightarrow \mathrm{GL}_1(\mathbf{C}) by

\rho_n(x) =  \bigl( \exp (2\pi i nx) \bigr).

(The right-hand side is a 1 \times 1-matrix.) If f : \mathbf{R} \rightarrow \mathbf{C} is a periodic function, then, according to Diaconis’ definition, the Fourier transform of f at \rho_{-n} is the 1 \times 1 matrix with entry

\int_0^{1} f(x) \exp(-2\pi i n x).

Up to scaling factors, this is one of the Fourier coefficients of f. The coefficient above measures the correlation of f with the function x \mapsto \exp(2\pi i n x). In the general setup, we can define, and measure, the correlation of a probability distribution on a finite group with a matrix representation of that group.

(2) To explain the group algebra, I’ll use the example where G is the symmetric group on set \{1,2,3\}. The group algebra \mathbf{C}G of G over \mathbf{C} consists of all formal \mathbf{C}-linear combinations of the elements of G. For example, one element of \mathbf{C}G is

(12) - \mathrm{i} (123) + (13).

The product in \mathbf{C}G comes from the group G, so for instance

(\mathrm{id} + (12))(\mathrm{id} - (123)) = \mathrm{id} + (12) - (123) - (23)

where \mathrm{id} denotes the identity element of G.

The interesting thing is that if we identity probability measures G \rightarrow \mathbf{C} with elements of \mathbf{C}G, then the product in the group algebra corresponds to the convolution product on probability measures

(P \star Q)(s) = \sum_{t \in G} P(st^{-1})Q(t).

Moreover under this identification, the Fourier transform with respect to a representation \rho : G \rightarrow \mathrm{GL}_n(\mathbf{C}) becomes an algebra homomorphism from \mathbf{C}G into the algebra of all n \times n-matrices. So, as we’d hope, the Fourier transform of a convolution of two functions is just the product of the Fourier transforms of each function. In symbols:

\widehat{P \star Q}(\rho) = \hat{P}(\rho) \hat{Q}(\rho).

The previous equation can be checked without reference to the group algebra in a few lines; still it’s nice that there is a decent algebraic explanation for why it holds.

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