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Cyclotomic Matrices and Power Difference Sets

Wei-Liang Sun

TL;DR

This work reinterprets cyclotomic numbers $(i,j)$ through the cyclotomic matrix $A=[(i,j)]$ and ties their algebraic structure to Schur rings, yielding new product identities and normality-like relations. It then leverages this framework to analyze the power difference set problem, providing equivalent matrix characterizations (e.g., $A^T A=\lambda k J_{\ell}+(k-\lambda)I_{\ell}-kE_{0,0}$) and explicit spectra and determinants for key matrices, thereby linking finite-field combinatorics with linear-algebraic structure. The paper extends the analysis to shifted and modified power-difference sets, derives their spectra and Gram-structure, and presents several necessary conditions and parity constraints that illuminate the existence landscape. Finally, it discusses open questions at the interface of cyclotomic-number theory, Schur rings, and geometric simplex constructions, suggesting directions for future work on when such difference sets can occur and how their algebraic structure might constrain their existence.

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.

Cyclotomic Matrices and Power Difference Sets

TL;DR

This work reinterprets cyclotomic numbers through the cyclotomic matrix and ties their algebraic structure to Schur rings, yielding new product identities and normality-like relations. It then leverages this framework to analyze the power difference set problem, providing equivalent matrix characterizations (e.g., ) and explicit spectra and determinants for key matrices, thereby linking finite-field combinatorics with linear-algebraic structure. The paper extends the analysis to shifted and modified power-difference sets, derives their spectra and Gram-structure, and presents several necessary conditions and parity constraints that illuminate the existence landscape. Finally, it discusses open questions at the interface of cyclotomic-number theory, Schur rings, and geometric simplex constructions, suggesting directions for future work on when such difference sets can occur and how their algebraic structure might constrain their existence.

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.

Paper Structure

This paper contains 11 sections, 32 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A connection to Schur rings
  4. Inner product of columns in cyclotomic matrices
  5. Power Difference Sets
  6. Some necessary conditions and equivalent statements
  7. Cyclotomic matrix to a power difference set
  8. Two additional properties
  9. Shifted cyclotomic matrices
  10. Modified power difference set
  11. Concluding Remark

Key Result

Lemma A

Denote $A_x = \left[ (i-x, j-x) \right]_{0 \leq i, j \leq \ell-1}$ for $x \in \mathbb{Z}$. For $0 \leq u, v \leq \ell-1$, one has where $I_{\ell}$ is the $\ell$-by-$\ell$ identity matrix, $\delta_{s,t}$ is defined to be $k$ if $s \equiv t \pmod{\ell}$ and $0$ otherwise, and $E_{s,t}$ is the all-zero matrix except for a $1$ in the $(s, t)$-entry.

Theorems & Definitions (63)

  • Lemma A
  • Conjecture
  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 3.1
  • ...and 53 more