The Computation of Fourier transforms on SL₂(ℤ/pⁿℤ) and related numerical experiments

2017/10/07 by Breen, Benjamin K., Deford, Daryl R., Linehan, Jason D. +1
#20C40 #43-04 #65T50 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1710.02687

Abstract

We detail an explicit construction of ordinary irreducible representations for the family of finite groups SL₂(\mathbb Z /pⁿ \mathbb Z) for odd primes p and n≥ 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 SL₂(\mathbb Z /p² \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 SL₂(\mathbb Z/p \mathbb Z) and suggesting conjectures for the expansion properties of such graphs.

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