Experimental Mathematics

Fast Fourier analysis for {${\rm SL}\sb 2$} over a finite field and related numerical experiments

John D. Lafferty and Daniel Rockmore

Abstract

We study the complexity of performing Fourier analysis for the group $\SL_2(\fq)$, where $\fq$ is the finite field of $q$ elements. Direct computation of a complete set of Fourier transforms for a complex-valued function $f$ on $\SL_2(\fq)$ requires $q^6$ operations. A similar bound holds for performing Fourier inversion. Here we show that for both operations this naive upper bound may be reduced to $O(q^4\log q)$, where the implied constant is universal, depending only on the complexity of the "classical'' fast Fourier transform. The techniques we use depend strongly on explicit constructions of matrix representations of the group.

Additionally, the ability to construct the matrix representations permits certain numerical experiments. By quite general methods, recent work of others has shown that certain families of Cayley graphs on $\SL_2(\fq)$ are expanders. However, little is known about their exact spectra. Computation of the relevant Fourier transform permits extensive numerical investigations of the spectra of these Cayley graphs. We explain the associated calculation and include illustrative figures.

Article information

Source
Experiment. Math., Volume 1, Issue 2 (1992), 115-139.

Dates
First available in Project Euclid: 26 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1048709049

Mathematical Reviews number (MathSciNet)
MR94f:11131

Zentralblatt MATH identifier
0773.65091

Subjects
Primary: 11Y16: Algorithms; complexity [See also 68Q25]

Citation

Lafferty, John D.; Rockmore, Daniel. Fast Fourier analysis for {${\rm SL}\sb 2$} over a finite field and related numerical experiments. Experiment. Math. 1 (1992), no. 2, 115--139. https://projecteuclid.org/euclid.em/1048709049


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