The efficient computation of Fourier transforms on semisimple algebras

TL;DR

This paper introduces a diagrammatic method for efficiently computing Fourier transforms on semisimple algebras, extending previous group-based algorithms to a broader class of algebras with practical applications.

Contribution

It develops a general approach using Bratteli diagrams and path algebras to create efficient Fourier transform algorithms for various semisimple algebras.

Findings

Developed algorithms for Brauer, Temperley-Lieb, and Birman-Murakami-Wenzl algebras.

Extended Fourier transform computation techniques beyond finite groups.

Achieved high efficiency in algebra-specific Fourier transform algorithms.

Abstract

We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path algebra and construction of Gel'fand-Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley-Lieb algebras, and Birman-Murakami-Wenzl algebras.