The Computation of Fourier transforms on $SL_2(\mathbb{Z}/p^n\mathbb{Z}) and related numerical experiments

Abstract

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $S{L}_2(\mathbb{Z}/p^n\mathbb{Z})$ for odd primes $p$ and $n\ge 2$. For $n=2$, the construction is a complete set of irreducible complex representations, while for $n>2$, all but a handful are obtained. We also produce an algorithm for the computation of a Fourier transform for a function on $S{L}_2(\mathbb{Z}/p^2\mathbb{Z})$. With this in hand we explore the spectrum of a collection of Cayley graphs on these groups, extending analogous computations for Cayley graphs on $S{L}_2(\mathbb{Z}/p\mathbb{Z})$ and suggesting conjectures for the expansion properties of such graphs.

New title, more detail in the constructions, 24 pages, 5 figures, 6 tables