Positivity of GCD tensors and their determinants
Projesh Nath Choudhury, Krushnachandra Panigrahy
TL;DR
This work studies positivity and determinants of GCD tensors. It proves that GCD tensors on a set of distinct positive integers are strongly completely positive and remain so under fractional Hadamard powers, with explicit tensor decompositions based on incidence matrices and Euler's totient. It then derives a closed-form determinant for factor-closed supports, det(T_{[S]})=\prod_i \Phi(s_i)^{(m-1)^{(n-1)}}, and extends determinant identities under multiplicative maps via g*\mu, as well as a generalized totient Ψ_S for GCD-closed and meet-lattice settings. The results are unified through a tensor-factorization framework and extend to meet tensors on lattices, yielding analogous determinant identities det(\mathcal{T}_{[S]}^{g})=\prod_i \Psi_{S,g}(s_i)^{(m-1)^{(n-1)}}.
Abstract
Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $T_{[S]}=(t_{i_{1}i_{2}\ldots i_{m}}),$ where $t_{i_{1}i_{2}\ldots i_{m}}=GCD(s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}),$ the greatest common divisor (GCD) of $s_{i_{1}},s_{i_{2}},\ldots,$ and $s_{i_{m}}$ is called the GCD tensor on $S$. The earliest result on GCD tensors goes back to Smith [Proc. Lond. Math. Soc., 1976], who computed the determinant of GCD matrix on $S=\{1,2,\ldots,n\}$ using the Euler's totient function, followed by Beslin-Ligh [Linear Algebra Appl., 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the $k$-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number $r,$ the tensor $T_{[S]}^{\circ r}=(t^{r}_{i_{1}i_{2}\ldots i_{m}})$ is strongly CP. Finally, we obtain an interesting decomposition of GCD tensors using Euler's totient function $Φ$. Using this decomposition, we show that the determinant (also called hyperdeterminant) of the $m$th-order GCD tensor $T_{[S]}$ on a factor-closed set $S=\{s_1,\dots,s_n\}$ is $\prod\limits_{i=1}^{n} Φ(s_{i})^{(m-1)^{(n-1)}}$.