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 $SL_2({\mathbb Z} /p^n {\mathbb Z})$ 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.02687
 arXiv:
 arXiv:1710.02687
 Bibcode:
 2017arXiv171002687B
 Keywords:

 Mathematics  Representation Theory;
 4304;
 65T50;
 20C40
 EPrint:
 New title, more detail in the constructions, 24 pages, 5 figures, 6 tables