Double Coset Decompositions and Computational Harmonic Analysis on Groups

In this paper we introduce new techniques for the eecient computation of a Fourier transform on a nite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the Cooley-Tukey FFT to arbitrary nite groups. We apply our general results to special linear groups and low rank symmetric groups, and obtain new eecient algorithms for harmonic analysis on these classes of groups, as well as the two-sphere. 1. Introduction The Fourier transform on a nite group is a generalization of the discrete Fourier transform of a nite data sequence. The analogues of the complex exponentials appearing in the discrete Fourier transform are the matrix elements of a complete set of irreducible complex matrix representations. The matrix elements form a basis for the space of complex valued functions on the group, and the Fourier transform describes the expansion of functions in this basis. From this point of view, the usual discrete Fourier transform is simply the Fourier transform on a cyclic group. In the current paper we study algorithms for the eecient computation of Fourier transforms on arbitrary nite groups. On an abelian group the Fourier transform may be computed using the fast Fourier transform algorithms of Cooley and Tukey 17], and their many variants (see e.g. 26]). Our results may be considered a generalization of the Cooley-Tukey algorithm to nonabelian nite groups. The computation of Fourier transforms on nonabelian nite groups was rst studied by Willsky 53] in the context of lter design. Since then, various authors have treated solvable groups 8, 12, 49], supersolvable groups 4, 5], symmetric groups 13, 14, 38] and their wreath products 48], special linear groups 36], and also the general case of arbitrary nite groups 14, 22, 41]. The book 15] and the survey article 40] present overviews of this subject. For a discussion of applications see 18, 19, 20] and the survey article 47]. The algorithms we develop in this paper are examples of the \separation of variables" approach developed in 41, 42] and 40]. In addition to our general theorems on the computation of Fourier transforms, we have focused on the example of the special linear groups. Fourier transforms on this group are of interest for certain applications to error correcting codes 35]. Our main concrete result for special linear groups is as follows.