Hyperdeterminantal Total Positivity

TL;DR

This paper introduces hyperdeterminantal total positivity, extending classical total positivity to higher dimensions, and constructs new examples of such kernels using advanced algebraic formulas and group theory.

Contribution

It defines hyperdeterminantal total positivity, proves positivity for a new exponential kernel, and generalizes key formulas and concepts using hyperdeterminants and reflection groups.

Findings

Established hyperdeterminantal total positivity for a specific exponential kernel

Generalized Karlin's composition formula via hyperdeterminantal Binet-Cauchy formula

Constructed multiple examples of hyperdeterminantal totally positive kernels

Abstract

For a given positive integer $m$, the concept of hyperdeterminantal total positivity is defined for a kernel $K\colon {\mathbb R}^{2m} \to {\mathbb R}$, 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 $K(x_1,\dots,x_{2m}) = \exp(x_1\cdots x_{2m})$, $x_1,\dots,x_{2m} \in \mathbb{R}$ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula; then we use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity by means of the theory of finite reflection groups are described and some open problems are posed.