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 be a finite group and let be a function. (In practice, is often a probability measure on , but we don’t need this for the definition to make sense.) Let be representation of ; that is, is a group homomorphism from into the group of invertible matrices. The Fourier transform of at the representation is
So the Fourier transform is a function from representations to matrices.
(1) All this works fine if we take , provided we replace the sum with an integral and require the functions to be periodic, say with period . (So really we are looking at representations of the unit circle.) For each we may define a representation by
(The right-hand side is a -matrix.) If is a periodic function, then, according to Diaconis’ definition, the Fourier transform of at is the matrix with entry
Up to scaling factors, this is one of the Fourier coefficients of . The coefficient above measures the correlation of with the function . 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 is the symmetric group on set . The group algebra of over consists of all formal -linear combinations of the elements of . For example, one element of is
The product in comes from the group , so for instance
where denotes the identity element of .
The interesting thing is that if we identity probability measures with elements of , then the product in the group algebra corresponds to the convolution product on probability measures
Moreover under this identification, the Fourier transform with respect to a representation becomes an algebra homomorphism from into the algebra of all -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:
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.