Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 055, 18 pages      arXiv:2412.03000      https://doi.org/10.3842/SIGMA.2025.055
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Hyperdeterminantal Total Positivity

Kenneth W. Johnson a and Donald St. P. Richards b
a) Department of Mathematics, Pennsylvania State University, Abington, Pennsylvania 19001, USA
b) Department of Statistics, Pennsylvania State University, University Park, PA 16802, USA

Received December 04, 2024, in final form July 03, 2025; Published online July 11, 2025

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.

Key words: Binet-Cauchy formula; determinant; generalized hypergeometric functions of matrix argument; Haar measure; hyperdeterminant; Schur function; unitary group; zonal polynomials.

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