In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...

Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform. This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions. The theory for applying the Generalized Fourier Transform is explained, building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform. The classical convolution theorem and diagonalization results are generalized to the non-commutative case of block diagonalizing equivariant matrices. Our presentation stresses the connection between multiplication with an equivariant matrices and the application of a convolution. This approach highlights the role of the underlying mathematical structures such as the group algebra, and it also simplifies the application of fast Generalized Fourier Transforms. The theory is illustrated with a selection of numerical examples. Key words: Non commutative Fourier analysis, equivariant operators, block diagonalization. 1 Introduction.

Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for high-performance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute using earlier techniques, including a win requiring a record 243 moves. (3) To contribute to the study of the history of chess endgames, by focusing on the work of Friedrich Amelung (in particular his apparently lost analysis of certain six-piece endgames) and that of Theodor Molien, one of the founders of modern group representation theory and the first person to have systematically numerically analyzed a pawnless endgame. 1.

Abstract. We present an improved, memory efficient retrograde algorithm we developed during our research on solving Chinese chess endgames. This domain-independent retrograde algorithm, along with a carefully designed domain-specific indexing function, has enabled us to solve many interesting Chinese chess endgame on standard consumer class hardware. We also report some of the most interesting results here. Some of these are real surprises for human Chinese chess experts. For example, the aegp-aaee 1 ending is a theoretical win, not as previously believed, a draw. Human analysis for this endgame over many years by top players has been proved to be wrong. 1.

5-15-2008 In mathematics you don’t understand things. You just get used to them. Johann von Neumann v Contents

We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees ” of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first is a characterization of minimal conjugate generating sets of Sn. The second is a generalization of a result due to Feng characterizing the automorphism groups of the Cayley graphs formed by certain generating sets composed of cycles. We compute the full automorphism groups subject to a weak condition and conjecture that the characterization still holds without the condition. We also present some computational results in relation to Hamiltonicity of Cayley graphs, including a generalization of the work on quasi-hamiltonicity by Gutin and Yeo to undirected graphs.