Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II

Abstract

Let a, b and h be positive integers and S = {x 1, … , x h } be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ of {1, … , h} such that x σ(1)|…|x σ(h). We say that the set S consists of two coprime divisor chains if we can partition S as S = S 1 ∪ S 2, where S 1 and S 2 are divisor chains and each element of S 1 is coprime to each element of S 2. The matrix having the ath power (x i , x j ) a of the greatest common divisor (GCD) of x i and x j as its (i,j)-entry is called the ath power GCD matrix defined on S, denoted by (S a ). Similarly, we can define the ath power least common multiple (LCM) matrix [S a ]. In the first paper of the series, Tan [Q. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659--671] showed that if S consists of two coprime divisor chains and 1 ∈ S and a|b, then (S a )|(S b ), [S a ]|[S b ] and (S a )|[S b ] hold in the ring M h (Z) of h × h matrices over integers. But such factorizations need not hold if a ∤ b. In this second paper of the series, we assume that S consists of two coprime divisor chains and 1 ∉ S. We show the following results: (i) If , then , and . (ii) If a|b, then det(S a ) | det(S b ), det[S a ] | det[S b ] and det(S a ) | det[S b ]. (iii) If a|b, then (S a ) | (S b ), [S a ] | [S b ] and (S a ) | [S b ] hold in the ring M h (Z) if and only if both and are integers, where S = S 1 ∪ S 2 with S 1 and S 2 divisor chains and x = min(S 1) and y = min(S 2). Our results extend Hong’s results and complement Tan’s results.