Adapted diameters and the efficient computation of Fourier transforms on finite groups

This paper introduces new techniques for the efficient computation of the Discrete Fourier Transform (DFT) for a finite group G and in so doing, relates the complexity of a finite group to its adapted diameter, relative to a given generating set and chain of subgroups. Consequently, we are able to show, for the first time, that the complexity of the DFT of a finite group is intimately related to group structure and thereby begin to link two major areas of research in computational group theory. In many particular cases, the resulting algorithms have potential applications for data analysis and signal processing. Given a chain of subgroups for a group G we introduce a technique which produces factorizations of group elements into short products of elements which commute with various subgroups along the chain. The commutativity properties of the factors is used to to show that for any irreducible matrix representation of the group these elements will have factorizations as highly structured sparse matrices. This allows a separation of variables style algorithm to be used in computing the associated DFT and consequent speedups follow immediately. In particular, this technique recovers the best known algorithms for the symmetric groups and wreath products and beyond that, dramatically improves on the known complexity of the DFT for the finite groups GL(n, q) and gives first complexity results for all finite classical groups, finite groups of Lie type and groups with a (B, N)-pair.