Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

TL;DR

This paper introduces a diagrammatic method for constructing efficient Fourier transform algorithms on finite groups, improving computational bounds for various groups and connecting algebraic structures to algorithm complexity.

Contribution

It extends diagrammatic and algebraic frameworks to develop more efficient Fourier transform algorithms for finite groups, including new bounds and generalizations.

Findings

Improved upper bounds for Fourier transforms on finite groups

Unified diagrammatic approach for various groups

Recovery of best known algorithms for symmetric and Lie groups

Abstract

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.