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The efficient computation of Fourier transforms on the symmetric group. (English) Zbl 0902.20005
The paper describes techniques for the computation of Fourier transforms on symmetric groups and their homogeneous spaces. In particular, the matrix multiplication of Clausen’s algorithm is replaced by sums indexed by combinatorial objects that generalize Young tableaux, which are written in a form similar to Horner’s rule. The resulting algorithm computes the Fourier transform of a function on S n by 3 4n(n-1)n! multiplications and the same number of additions. The corresponding results for the inverse transforms and transforms on homogeneous spaces are also included.
MSC:
20C40Computational methods (representations of groups)
20C30Representations of finite symmetric groups
65T50Discrete and fast Fourier transforms (numerical methods)
05E10Combinatorial aspects of representation theory