Generating fast Fourier transforms of solvable groups

This paper presents a new algorithm for constructing a complete list of pairwise inequivalent ordinary irreducible representations of a finite solvable group G. The input of the algorithm is a pc-presentation corresponding to a composition series refining a chief series of G. Modifying the Baum-Clausen-Algorithm for supersolvable groups and combining this with an idea of Plesken for constructing intertwining spaces, we derive a worst-case upper complexity bound O(p · |G| log(|G|)), where p is the largest prime divisor of |G|. The output of the algorithm is well-suited to perform a fast Fourier transform of G. For supersolvable groups there are composition series which are already a chief series. In this case the generation of DFTs can be done more efficiently than in the solvable case. We report on a recent implementation for the class of supersolvable groups.