Abstracts

Positivity of GCD tensors and their determinants

Krushnachandra Panigrahy ,  NIT Agartala

Let $S = \{ s_1, s_2, \dots, s_n \}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $\mathcal{T}_{[S]} = (t_{i_1 i_2 \cdots i_m})$ – where $t_{i_1 i_2\cdots i_m} = {\rm GCD}(s_{i_1}, s_{i_2}, \dots, s_{i_m})$ is the greatest common divisor of $s_{i_1}, \dots, s_{i_m}$ – is called the GCD tensor on $S$. By a surprising result of Beslin and Ligh (1989), all GCD matrices are positive definite. In my talk, (i) I will discuss infinite divisibility and complete positivity of higher-order tensors. (ii) I will present a formula for the determinant (also called hyperdeterminant) of the $m$th-order GCD tensor in terms of Euler's totient function. (This is a joint work with Projesh Nath Choudhury.)