Hyperdeterminantal Total Positivity
Kenneth W. Johnson, Donald St. P. Richards

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 , the concept of hyperdeterminantal total positivity is defined for a kernel , thereby generalizing the classical concept of total positivity. Extending the fundamental example, , , of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel , 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…
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Taxonomy
TopicsQuantum Mechanics and Applications
