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

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $S{L}_2(\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z})$ 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 $S{L}_2(\mathbb{Z}\mathrm{/}{p}^2\mathbb{Z})$. With this in hand we explore the spectrum of a collection of Cayley graphs on these groups, extending analogous computations for Cayley graphs on $SL_2({\mathbb Z}/p {\mathbb Z})$ and suggesting conjectures for the expansion properties of such graphs.