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This thesis presents the results of an investigation into the representation theory of the alternating and dihedral groups and explores how their irreducible representations can be distinguished with the use of class sums.

In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research.

A discrete Fourier transform, or DFT, is an isomorphism from a group algebra to a direct sum of matrix algebras. An algorithm that efficiently applies a DFT is called a fast Fourier transform, or FFT. The concept of a DFT will be introduced and examined from both a general and algebraic perspective. We will then present and analyze a specific FFT for the symmetric group.

Thesis--Ph. D., University of Hong Kong, 1986.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (M.Sc.)--University of Melbourne, [Dept. of Mathematics and Statistics], 2009. MSc (by Research) / Typescript. Includes bibliographical references.

Thesis (Ph. D.)--West Virginia University, 2000. / Title from document title page. Document formatted into pages; contains iii, 56 p. Includes abstract. Includes bibliographical references (p. 54-56).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2004. / Includes bibliographical references (p. 123-126).

First published in the Bulletin of the American Mathematical Society in Vol.69, 1963, published by the American Mathematical Society