David Maslan ; Daniel N. Rockmore ; Sarah Wolff

Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
dmtcs:6372 
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

https://doi.org/10.46298/dmtcs.6372
Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
Authors: David Maslan ^{1}; Daniel N. Rockmore ^{2}; Sarah Wolff ^{3}
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David Maslan;Daniel N. Rockmore;Sarah Wolff
1 HBK Capital Management
2 Department of Mathematics [Dartmouth]
3 Department of Mathematics and Computer Science [Granville]
We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.