Hyperdeterminantal Total Positivity
For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| ISSN: | 1815-0659 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/213521 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel (₁, …, ₂ₘ) = exp(₁⋯₂ₘ), ₁,…,₂ₘ ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula. We use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity via the theory of finite reflection groups are described, and some open problems are posed.
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| ISSN: | 1815-0659 |