Fast Fourier Analysis for SL 2 over a Finite Fieldand Related

This paper studies the complexity of performing Fourier analysis for the group SL 2 (K) where K is the nite eld of q elements. Direct computation of a complete set of Fourier transforms for a complex-valued function f on SL 2 (K) 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 here is universal, depending only on the complexity of the \classi-cal" FFT. The techniques presented here 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 (K) 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 to be carried out. We explain the associated calculation and include gures which detail our results. Fast Fourier analysis for SL 2 over a nite eld 2 1 Introduction