  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.
