A Decimation-in-frequency Fast Fourier Transform for the Symmetric Group

1. A Decimation-in-frequency Fast Fourier Transform for the Symmetric Group Masanori Koyama Thesis advisor : Michael E.Orrison Operation counts Orrison-DIF Let K, H be subgroups of G. Double coset frequency space(DCF) is a subspace of that results from decimation of in the frequency domain followed by decimation of the resulting spaces by (K,H) double coset spaces . The Orrison-decimation-in-frequency (ODIF) is a promising type of FFT that computes the Fourier coefficients in steps by a series of local change of bases in DCFs. Discrete Fourier Transform A Discrete Fourier Transform (DFT) is a change of basis of a group algebra from the standard basis (consisting of group elements) to a basis that respects the ‘s decomposition into spaces that are irreducible under the regular action of (Fourier basis). For example, decomposes into four irreducible spaces under the regular action: A Fast Fourier Transform (FFT) is an efficient application of a DFT. Coordinates in the Fourier basis are Fourier coefficients. A group algebra in the standard basis is a group algebra in the time domain, and the group algebra in a Fourier basis is a group algebra in the frequency domain. Problem Compute the operation count of ODIF for . • Other interesting results • Theorem: DIF and DIT are module theoretically equivalent. • Theorem: If is a Fourier basis of that respects the decomposition of into both left and right irreducible modules for any , and if is a Fourier basis of that respects the decomposition of into both left and right irreducible modules for any , then is a spanning set of that respects its decomposition into irreducible bimodules for all , . • Conjecture: The set of nonzero elements in is an orthogonal basis of . We call the this basis a tensor-basis of . Properties of double coset spaces Theorem: Any pair of double coset spaces of the same dimension are isomorphic as bimodules.We say is of type if . Theorem: If denotes the number of double cosets of type m in the decomposition of a double coset of type , then Theorem : The representation theory of double coset spaces can be computed recursively. DIT and DIF Decimation-in-frequency (DIF) is a method of FFT that divides up the computation of the Fourier coefficients by decimating the group algebra in the frequency domain. Decimation-in-time (DIT) is a method of FFT that divides up the computation of the Fourier coefficients by decimating the group algebra in the time domain. • Acknowledgements: • I would like to thank my advisor, Professor Michael Orrison, for his support and guidance. I would also like to thank Mike Hansen for providing me with code. Comparison of operation counts for ODIF full bound, ODIF implemented with tensor basis, and Naïve implementation of DIT.