The efficient computation of Fourier transforms on the symmetric group
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- by David K. Maslen PDF
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Abstract:
This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen’s algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the result in a form similar to Horner’s rule. The algorithm we obtain computes the Fourier transform of a function on $S_n$ in no more than $\frac {3}{4} n(n-1) |S_n|$ multiplications and the same number of additions. Analysis of our algorithm leads to several combinatorial problems that generalize path counting. We prove corresponding results for inverse transforms and transforms on homogeneous spaces.References
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Additional Information
- David K. Maslen
- Affiliation: Institut des Haute Études Scientifiques, Le Bois-Marie, 35 Route de Chartres, 91440, Bures-sur-Yvette, France
- Address at time of publication: Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098SJ, Amsterdam, The Netherlands
- Email: maslen@cwi.nl
- Received by editor(s): August 21, 1996
- Received by editor(s) in revised form: April 23, 1997
- Additional Notes: An extended abstract summarizing these results appears in the FPSAC ’97 proceedings volume.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp. 67 (1998), 1121-1147
- MSC (1991): Primary 20C30, 20C40; Secondary 65T20, 05E10
- DOI: https://doi.org/10.1090/S0025-5718-98-00964-8
- MathSciNet review: 1468943