Some lower and upper complexity bounds for generalized Fourier transforms and their inverses
SIAM Journal on Computing
Computing irreducible representations of supersolvable groups
Mathematics of Computation
Computing character tables of p-groups
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Computing irreducible representations of supersolvable groups over small finite fields
Mathematics of Computation
Fast multiplication and growth in groups
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Algebraic Complexity Theory
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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: CG ≃ ⊕kh=1Cdk×dk. Every such isomorphism D, a so-called discrete Fourier transform of CG, consists of a full set of pairwise inequivalent irreducible representations Dk 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(n log n) operations. Thus in this model, every O(n log n) 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 = ⊕Dk of CG and generates an O(n log n) FFT of CG. The running time to construct D is essentially proportional to the time to write down all the monomial (!) twiddle factors Dk(gi) where the gi are the generators corresponding to the pc-presentation. Finally, we sketch some applications.