What is a true spectra of a finite Fourier transform

TL;DR

This paper investigates the spectral properties of a modified finite Fourier transform on finite abelian groups, revealing that its squared form permutes elements iff the underlying bijection is a group isomorphism, and explores how spectra depend on this isomorphism.

Contribution

It characterizes when the squared modified Fourier transform acts as a permutation and analyzes how the spectrum varies with different group isomorphisms.

Findings

$( ilde{j} ext{F})^2$ is a permutation iff $j$ is a group isomorphism

Spectral properties depend on the choice of the isomorphism $j$

Provides conditions linking Fourier transform permutations and group isomorphisms

Abstract

In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$ is defined by the composition with a bijection $j:G\to \hat{G}$. ($\tilde j$ is a pullback of $j$.) In particular, we show that $(\tilde{j}\circ {\mathcal F})^2$ is a permutation if and only if $j$ is a group isomorphism. Then, we study how the spectra of $\tilde{j}\circ{\mathcal F}$ depends on the isomorphism $j$.