A representation-theoretic approach to the DFT with noncommutative generalizations

It is known that both the one-dimensional and multidimensional DFTs (discrete Fourier transforms) can be constructed as transition matrices associated with the decomposition of finite-dimensional complex commutative group algebras into simple components. Two key attributes of these transforms, orthogonality and the convolution property, are inherent in such a description, suggesting the possibility of enlarging the class by extending the construction to noncommutative groups. In this context, one speaks of a noncommutative or generalized transform, the definition of which is based on the theory of semisimple rings. The author reviews the ring theory and representation theory fundamental to the existence and computation of group algebra decompositions and sketches the representation-theoretic construction of both the classical and noncommutative discrete Fourier transforms. The noncommutative transform associated with the class of dihedral groups is explicitly constructed and shown directly to exhibit both orthogonality and a noncommutative convolution property.<>