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

TL;DR

This paper constructs explicit irreducible representations for the groups SL_2 over modular integers, develops an algorithm for Fourier transforms on these groups, and investigates their Cayley graph spectra to explore expansion properties.

Contribution

It provides a detailed construction of irreducible representations for SL_2(Z/p^nZ) and an algorithm for Fourier transforms on these groups, extending previous work.

Findings

Complete irreducible representations for n=2

Algorithm for Fourier transform on SL_2(Z/p^2Z)

Spectral analysis of Cayley graphs suggesting expansion conjectures

Abstract

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