Alternating Power Difference and Matrix Symmetry: Closed-Form Formulas for the First Appearance Degree $m_1$

TL;DR

This paper investigates the algebraic behavior of the Alternating Power Difference (APD) and its first appearance degree for specific matrix classes, deriving closed-form formulas and conjectures that reveal deep structural relationships.

Contribution

It introduces new formulas and conjectures for the first appearance degree of APD in various matrices, advancing understanding of matrix structure and combinatorial properties.

Findings

Closed-form formulas for the first appearance degree $m_1(A)$.

APD remains zero until a specific degree, showing a first appearance phenomenon.

Conjectures applicable across multiple matrix classes.

Abstract

This paper focuses on an integer-valued function $f_A(\sigma) := \operatorname{tr}(A P_\sigma)$ defined uniformly from a specific square matrix $A$ of order $n$ and a permutation $\sigma$ on the symmetric group $S_n$. The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as $APD_m(f_A)$, and its first appearance degree $m_1(f_A)$ for this function $f_A$ across various matrix classes. Specifically, we address special matrices such as shifted $r$-th power lattices, Vandermonde matrices, and circulant matrices, analyzing the phenomenon where the value of $APD_m(A)$ remains zero as $m$ increases until a specific degree (the first appearance phenomenon). In particular, we explore closed-form formulas for the first appearance degree $m_1(A)$ and the first appearance value $APD_{m_1}(A)$, presenting Conjectures that hold across multiple matrix classes. These results suggest a deep relationship between the structure of matrices and the analytical properties of functions on the symmetric group, providing new perspectives in matrix theory and combinatorics.