Honors Project On-Campus Access Only
The discrete Fourier transform on a group G converts data associated with group elements to a basis on matrix representations of G. This algorithm was famously made more efficient by the Cooley-Tukey fast Fourier transform (FFT), which has also been extended to groups. Two essential components of the FFT on groups are the efficient precomputation of matrix representations and the efficient factorization of group elements. Using a known factorization for the symmetric group as a model, we extend the factorization to two group-like algebraic structures: the Temperley-Lieb algebra and the Brauer algebra. The precomputation step is known for these algebras. Our main result is to provide an efficient factorization of a basis of each of these algebras into products of generators. This allows one to extend the FFT to both the Temperley-Lieb and Brauer algebras.
Datta, Esha, "The Efficient Factorization of Diagram Algebras" (2017). Mathematics, Statistics, and Computer Science Honors Projects. 59.
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Advisor: Tom Halverson