Cyclotomic Matrices and Power Difference Sets

TL;DR

This paper explores the algebraic structure of cyclotomic matrices, relating them to Schur rings, and applies this framework to analyze power difference sets through spectral and determinant conditions.

Contribution

It introduces a novel algebraic perspective on cyclotomic matrices and connects this to the power difference set problem, providing new structural identities and characterizations.

Findings

Revealed algebraic structures of cyclotomic matrices

Established conditions for power difference sets using spectral properties

Provided determinant characterizations related to cyclotomic matrices

Abstract

The cyclotomic matrix is commonly used to arrange cyclotomic numbers in a convenient format. A natural question is whether the structure of the matrix can reflect properties of these numbers. In this article, we examine cyclotomic numbers through their associated cyclotomic matrix and reveal an algebraic structure by relating it to a basis element of a Schur ring. This viewpoint leads to structural identities and reinterpretations of classical results. As an application, we investigate the power difference set problem and establish conditions expressed through cyclotomic matrices, including spectral and determinant characterizations.