Fourier analysis on finite groups

Choose one unitary representative of each irreducible complex representation. For a complex function on the finite group, sum its value times the representing matrix. The result is one square matrix per irreducible type. The number of scalar matrix entries is the sum of squared irreducible dimensions, exactly the group order, so no degrees of freedom are missing. Unitarity follows from an averaged Hermitian form; it is needed for the clean Hilbert–Schmidt energy formula, while the algebraic transform itself works in any fixed bases.

Define convolution by summing over a first group factor, with the second function evaluated at the inverse of that factor times the target element. Substituting a product for the target element in the Fourier sum factors the two independent sums in their original order. The convolution transform is therefore the first Fourier matrix multiplied by the second. For nonabelian groups these matrices need not commute, and neither does convolution. A scalar transform using only one-dimensional characters would lose the higher-dimensional blocks and cannot recover arbitrary functions.

Multiply a Fourier block by the representing matrix of the inverse of the desired group element, take its trace, weight by the irreducible dimension, and sum the results divided by the group order. Expanding the forward transform leaves a weighted sum of irreducible characters at the product of the inverse target and the summation element. That weighted sum is the regular character, which vanishes except when those two elements agree. The surviving term recovers the original function value exactly.

This argument proves inversion using finite sums, with no convergence or smoothness hypotheses. It also explains why the dimension weight and the factor of group order cannot be omitted. In the cyclic case every block is scalar and our chosen forward exponent has positive sign, so inversion has negative sign. Another common convention uses the opposite signs. Either can work, but a nonabelian inverse in the forward matrix formula also affects product order under a fixed convolution definition; formulas must be transported together.

Matrix-element orthogonality gives the energy identity, often called Plancherel or Parseval. Its weights differ from treating every matrix block as one scalar: the irreducible dimension multiplies the squared Hilbert–Schmidt norm. The identity offers a sensitive normalization check for numerical implementations. For a delta function at the identity every forward block is identity, and its weighted energy sum becomes the sum of squared irreducible dimensions divided by the group order, correctly producing one.

If the original function is constant on conjugacy classes, each Fourier block commutes with its irreducible group action and hence is scalar by Schur's lemma. This reduces convolution by central functions to scalar multiplication on each irreducible block. For a probability measure, the trivial block is its total mass, equal to one, while other blocks describe how repeated convolution changes nonconstant information. This interpretation organizes random walks on finite groups, although convergence still requires conditions such as irreducibility and aperiodicity rather than following from a Fourier transform alone.