A Fast Program Generator of Fast Fourier Transforms

Abstract

Let G be a finite group of order n. By Wedderburn’s Theorem, the complex group algebra ℂG is isomorphic to an algebra of block diagonal matrices: $\mathbb{C}G \simeq \oplus _{k = 1}^h \mathbb{C}^{d_k \times d_k }$. Every such isomorphism D, a so-called discrete Fourier transform of ℂG, consists of a full set of pairwise inequivalent irreducible representations D^k of ℂG. A result of Morgenstern combined with the well-known Schur relations in representation theory show that (under mild conditions) any straight line program for evaluating a DFT needs at least Ω(nlogn) operations. Thus in this model, every O(nlogn) FFT of ℂG is optimal up to a constant factor. For the class of supersolvable groups we will discuss a program that from a pc-presentation of G constructs a DFT D = ⊕D _k of ℂG and generates an O(nlogn) FFT of ℂG. The running time to construct D is essentially proportional to the time to write down all the monomial (!) twiddle factors D _k(g _i ) where the gi are the generators corresponding to the pc-presentation. Finally, we sketch some applications.