Halidon Rings and their Applications

TL;DR

This paper explores the computational properties and applications of Halidon rings, including their role in representation theory, Fourier transforms, and the development of computer codes for their analysis.

Contribution

It introduces Halidon rings as a new class of rings, examines their algebraic properties, and demonstrates their applications in Fourier analysis and representation theory with computational tools.

Findings

Halidon rings can be constructed beyond fields.

Computer codes for computing units, involutions, and idempotents in Halidon rings are developed.

Halidon rings can be used in Discrete Fourier Transform calculations.

Abstract

Halidon rings are rings with a unit element, containing a primitive $m^{th}$ root of unity and $m$ is invertible in the ring. The field of complex numbers is a halidon ring with any index $ m \geq 1$. In this article, the author examines the computational aspects of a new class of rings called Halidon rings and their applications with the help of computer codes. In representation theory, Maschke's theorem has an important role in studying the irreducible subrepresentations of a given group representation. Mainly the study is related to a finite field of characteristic which does not divide the order of the given finite group or the field of real or complex numbers. This article also examines the possibility of replacing the finite field in the theorem with halidon rings in such a way that the theorem is still valid. Halidon rings are rings with the unit element, containing a primitive $m^{th}$ root of unity and $m$ is invertible in the ring. The field of complex numbers is a halidon ring with any index $ m \geq 1$. Some computer codes have been developed to establish the existence of halidon rings which are not fields and the computation of units, involutions and idempotents in both halidon rings and halidon group rings. The application halidon rings in Discrete Fourier Transform (DFT) is studied and two computer codes have been developed to calculate DFT and inverse DFT