Hyperdeterminantal Total Positivity

Abstract: For a given positive integer $m$, the concept of {\it hyperdeterminantal total positivity} is defined for a kernel $K:\R^{2m} \to \R$, thereby generalizing the classical concept of total positivity. Extending the fundamental example, $K(x,y) = \exp(xy)$, $x, y \in \mathbb{R}$, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel $K(x_1,\ldots,x_{2m}) = \exp(x_1\cdots x_{2m})$, $x_1,\ldots,x_{2m} \in \mathbb{R}$ is established. A hyperdeterminantal Binet-Cauchy argument is applied to derive a generalization of Karlin's Basic Composition Formula, from which several examples of hyperdeterminantal totally positive kernels are constructed. Further generalizations of hyperdeterminantal total positivity by means of the theory of finite reflection groups are described and some open problems are posed.