Abstract

This paper presents a new algorithm for constructing a complete list of pairwise inequivalent ordinary irreducible representations of a finite solvable group G. The input of the algorithm is a pc presentation corresponding to a composition series refining a chief series of G. Modifying the Baum–Clausen algorithm for supersolvable groups and combining this with an idea of Plesken for constructing intertwining spaces, we derive a worst-case upper complexity bound O(p·|G|²log(|G|)), where p is the largest prime divisor of |G|. The output of the algorithm is well suited to performing a fast Fourier transform of G. For supersolvable groups there are composition series which are already chief series. In this case the generation of discrete Fourier transforms can be done more efficiently than in the solvable case. We report on a recent implementation for the class of supersolvable groups.

Author Keywords: Fast Fourier transform (FFT); Discrete Fourier transform (DFT)

Article Outline

1. Introduction

2. Background from representation theory

3. Basics for DFT generation of solvable groups

4. Algorithm M and complexity bounds

4.1. Algorithm RBC

4.2. Algorithm ET

4.3. Algorithm M

4.4. Analysis of Algorithm M

5. Implementation for supersolvable groups

6. Final remarks and future work

Acknowledgements

Appendix A

Appendix B

References