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Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
We present a general diagrammatic approach to the construction of efficient
algorithms for computing the Fourier transform of a function on a finite group.
By extending work which connects Bratteli diagrams to the construction of Fast
Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path
algebra connection to the construction of Gel'fand-Tsetlin bases and work in
the setting of quivers. We relate this framework to the construction of a {\em
configuration space} derived from a Bratteli diagram. In this setting the
complexity of an algorithm for computing a Fourier transform reduces to the
calculation of the dimension of the associated configuration space. Our methods
give improved upper bounds for computing the Fourier transform for the general
linear groups over finite fields, the classical Weyl groups, and homogeneous
spaces of finite groups, while also recovering the best known algorithms for
the symmetric group and compact Lie groups.Comment: 53 pages, 5 appendices, 34 figure
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