Fast Fourier Transforms for Wreath Products

In this paper fast Fourier transform algorithms (FFT's) are constructed for wreath products of the form GS n ]. Examples of interest include the hyperoctahedral groups B n (the symmetry groups of the n-cube) as well as more generally AS n ] for any abelian groups A and also the full wreath products S m S n ] and their iterates, often identiied as automorphism groups of spherically homogeneous rooted trees. In general, direct computation of the discrete Fourier transform for any nite group H requires jH j 2 operations. The algorithms presented in this paper provide substantial speed-ups for general wreath products GS n ], which for the particular examples mentioned above reduce the jH j 2 bound to O(jHj log 4 jH j). Thus new classes of nite groups of close to minimal linear complexity are obtained. Applications to data analysis, signal processing and molecular spectroscopy are discussed.