Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains
Let \(a\) be a positive integer and let \(S=\{x_1, x_2,\ldots, x_n\}\) be a set of distinct positive integers. The \(n\times n\) matrix whose \(ij\) entry is the \(a\)th power of the greatest common divisor of \(x_i\) and \(x_j\) is denoted by \((S^a)\) and is referred to as a power GCD matrix on \(S\). Power LCM matrices \([S^a]\) on \(S\) are defined similarly. The set \(S\) is said to consist of finitely many coprime divisor chains if \(S\) can be partitioned as \(S=S_1\cup S_2\cup\cdots\cup S_k\), where each \(S_i\) is a divisor chain and each element of \(S_i\) is coprime to each element of \(S_j\) for \(i\neq j\). The present authors show that if \(S\) consists of finitely many coprime divisor chains with \(1\in S\) and \(a\mid b\), then \((S^a)\mid (S^b)\), \([S^a]\mid [S^b]\) and \((S^a)\mid [S^b]\) in the ring of \(n\times n\) matrices over the integers and that if \(a\nmid b\), then such divisibility relations do not hold. This result confirms partially Conjectures 4.2--4.4 of \textit{S. Hong} [Linear Algebra Appl. 428, No. 4, 1001--1008 (2008; Zbl 1137.11017)]. The author also consider similar questions for finitely many ``quasi-coprime'' divisor chains.
scientific article; zbMATH DE number 6125878
scientific article