Efficient Computation of the Fourier Transform on Finite Groups
نویسنده
چکیده
Let G be a finite group, f: G --+ C a function, and p an ir-reducible representation of G. The Fourier transform is defined as j(p) =E SEG f(s)p(s) . Direct computation for all irreducible representations involvesorder IGI2 operations. We derive fast algorithms and develop them for thesymmetric groupSn. There, (n!)2 is reduced to n(n!)a/2, where a is theconstant for matrix multiplication (2.38 as of this writing). Variations of thealgorithm allow efficient computation for "small" representations. A practicalversion of the algorithm is given on Sn. Numerical evidence is presented toshow a speedup by a factor of 100 for n = 9 . DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138 DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YORK, NEW YORK, 10027 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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