Fourier transform on finite groups


In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.

Definitions

The Fourier transform of a function
at a representation of is
For each representation of, is a matrix, where is the degree of.
The inverse Fourier transform at an element of is given by

Properties

Transform of a convolution

The convolution of two functions is defined as
The Fourier transform of a convolution at any representation of is given by

Plancherel formula

For functions, the Plancherel formula states
where are the irreducible representations of

Fourier transform for finite abelian groups

If the group G is a finite abelian group, the situation simplifies considerably:
The Fourier transform of a function is the function given by
The inverse Fourier transform is then given by
For, a choice of a primitive n-th root of unity yields an isomorphism
given by. In the literature, the common choice is, which explains the formula given in the article about the discrete Fourier transform. However, such an isomorphism is not canonical, similarly to the situation that a finite-dimensional vector space is isomorphic to its dual, but giving an isomorphism requires choosing a basis.
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply where 0 is the group identity and is the Kronecker delta.
Fourier Transform can also be done on cosets of a group.

Relationship with representation theory

There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups. The set of complex-valued functions on a finite group,, together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of over the complex numbers,. Modules of this ring are the same thing as representations. Maschke's theorem implies that is a semisimple ring, so by the Artin–Wedderburn theorem it decomposes as a direct product of matrix rings. The Fourier transform on finite groups explicitly exhibits this decomposition, with a matrix ring of dimension for each irreducible representation.
More specifically, the Peter-Weyl theorem states that there is an isomorphism
given by
The left hand side is the group algebra of G. The direct sum is over a complete set of inequivalent irreducible G-representations.
The Fourier transform for a finite group is just this isomorphism. The product formula mentioned above is equivalent to saying that this map is a ring isomorphism.

Applications

This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized quickly using the fast Fourier transform, and this yields a fast method for solving systems of linear equations with circulant matrices. Similarly, the Fourier transform on arbitrary groups can be used to give fast algorithms for matrices with other symmetries. These algorithms can be used for the construction of numerical methods for solving partial differential equations that preserve the symmetries of the equations.