The Computation of Fourier Transforms on the Symmetric Group

This paper introducesnew techniques for the eecient computationof 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 Sn in no more than 3 4 n(n ? 1)jSnj multiplications and the same number of additions. Analysis of our algorithm leads to several enumerative combinatorial problems that generalize path counting. We describe corresponding results for inverse transforms and transforms on homogeneous spaces. 1. Introduction The harmonic analysis of a complex function on a nite cyclic group is the expansion of that function in a basis of complex exponential functions. This is equivalent to the discrete Fourier transform of a nite data sequence, and may be computed eeciently using the fast Fourier transform algorithms of Cooley and Tukey 6] or their many variants (see e.g. 11]). In the current paper we study the harmonic analysis of a function on the symmetric group. The analogs of the complex exponentials are the matrix entries of a complete set of irreducible complex matrix representations of S n , called matrix coeecients, and the expansion of functions in this basis may be computed by a generalized Fourier transform on the symmetric group. We describe eecient algorithms for computing the harmonic analysis of a function on the symmetric group, or equivalently, its generalized Fourier transform. Thus our results may be considered a generalization of the fast Fourier transform to the symmetric group. Fourier transforms on nite groups have been studied by many authors. The books of Beth 1], Clausen and Baum 2], and the survey article 18] are general references for the computational aspects of these transforms. Rockmore 21] and Diaconis 8] contain discussions of the applications. For applications more speciic to symmetric groups, see 7] and 10]. The computation of Fourier transforms on symmetric groups was rst studied by Clausen 4] 5], and Diaconis and Rockmore 9], using approaches related to the one taken in the current paper; also see 3] for a detailed discussion of Clausen's algorithm and its implementation. Linton, Michler, and Olsson 14] use a diierent method that involves the decomposition of Fourier transforms taken at a monomial representations. The algorithms we develop in the current paper are reenements of Clausen's algorithm 4] …