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Efficient Computation of the Fourier Transform on Finite Groups

Persi Diaconis and Daniel Rockmore
Journal of the American Mathematical Society
Vol. 3, No. 2 (Apr., 1990), pp. 297-332
DOI: 10.2307/1990955
Stable URL: http://www.jstor.org/stable/1990955
Page Count: 36
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Abstract

Let G be a finite group, f: G → C a function, and ρ an irreducible representation of G. The Fourier transform is defined as $\hat f(\rho) = \sum_{s \epsilon G}f(s) \rho(s)$. Direct computation for all irreducible representation involves order ∣ G ∣2 operations. We derive fast algorithms and develop them for the symmetric group Sn. There, (n!)2 is reduced to n(n!)a/2, where α is the constant for matrix multiplication (2.38 as of this writing). Variations of the algorithm allow efficient computation for "small" representations. A practical version of the algorithm is given on Sn. Numerical evidence is presented to show a speedup by a factor of 100 for n = 9.

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