Fast Fourier Analysis for SL2 over a Finite Field and Related Numerical Experiments

AMS Subject Classi cation: 20-04, 05C25, 20C30 Rockmore was supported in part by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship. We study the complexity of performing Fourier analysis for the group SL2(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 SL2(Fq) requires q6 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(q4 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.