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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Efficient computation of the Fourier transform on finite groups
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by Persi Diaconis and Daniel Rockmore
J. Amer. Math. Soc. 3 (1990), 297-332


Let $G$ be a finite group, $f:G \to {\mathbf {C}}$ a function, and $\rho$ an irreducible representation of $G$. The Fourier transform is defined as $\hat f(\rho ) = {\Sigma _{s \in G}}f(s)\rho (s)$. Direct computation for all irreducible representations involves order ${\left | G \right |^2}$ operations. We derive fast algorithms and develop them for the symmetric group ${S_n}$. There, ${(n!)^2}$ is reduced to $n{(n!)^{a/2}}$, where $a$ 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 ${S_n}$. Numerical evidence is presented to show a speedup by a factor of 100 for $n = 9$.
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Bibliographic Information
  • © Copyright 1990 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 3 (1990), 297-332
  • MSC: Primary 20C40; Secondary 20B40, 20C30, 68Q25
  • DOI:
  • MathSciNet review: 1030655