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Generalized Fourier transforms and their applications / by Sarah Wolff

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Author	Wolff, Sarah E.
Title	Generalized Fourier transforms and their applications / by Sarah Wolff
Date	2015
Abstract	This thesis centers around a generalization of the classical discrete Fourier transform. We first present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group or semisimple algebra. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms [65], we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of general semisimple algebras and quivers. We relate this framework to the construction of a 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. We give explicit counting results to find the dimension of these configuration spaces, and thus the complexity of the associated Fourier transform. Our methods give improved upper bounds 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. We extend these results further to semisimple algebras, giving the first results for non-trivial upper bounds for computing Fourier transforms on the Brauer and Birman-Murakami-Wenzl (BMW) algebras. The extension of our algorithm to Fourier transforms on semisimple algebras is motivated by emerging applications of such transforms. In particular, Fourier transforms on the Iwahori-Hecke algebras have been used to analyze Metropolis-based systematic scanning strategies for generating Coxeter group elements [25]. We consider the Metropolis algorithm in the context of the Brauer and BMW monoids and provide systematic scanning strategies for generating monoid elements. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We translate these walks into left multiplication operators in the corresponding BMW algebra. Taking this algebraic perspective enables the use of tools from representation theory to analyze the walks; in particular, we develop a norm arising from a trace function on the BMW algebra to analyze the time to stationarity of the walks.
Description	1 online resource (vii, 192 leaves) : illustrations
Format	application/pdf
College	Dartmouth College -- Department of Mathematics.
Issuing Body	Dartmouth Dissertations are made available through the Dartmouth Digital Library.
Degree	Thesis (Ph.D.)--Dartmouth College.
Copyright	All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means without permission in writing from the author.
DOI	http://dx.doi.org/10.1349/ddlp.1625

Title	Page 1
Full Text	GENERALIZED FOURIER TRANSFORMS AND THEIR APPLICATIONS A Thesis Submitted to the Faculty in partial ful llment of the requirements for the degree of Doctor of Philosophy in Mathematics by Sarah Wol DARTMOUTH COLLEGE Hanover, New Hampshire May 2015 Examining Committee: Daniel Rockmore, Chair Thomas Shemanske Sergi Elizalde Richard Foote F. Jon Kull, Ph.D. Dean of Graduate Studies

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