Hyperdeterminantal Total Positivity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916837823873024 |
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| author | Johnson, Kenneth W. Richards, Donald St. P. |
| author_facet | Johnson, Kenneth W. Richards, Donald St. P. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperdeterminantal Total Positivity Johnson, Kenneth W. Richards, Donald St. P. Classical Analysis and ODEs Statistics Theory Primary 33C20, 05E05, Secondary 15A15, 15A72, 33C80 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. |
| title | Hyperdeterminantal Total Positivity |
| topic | Classical Analysis and ODEs Statistics Theory Primary 33C20, 05E05, Secondary 15A15, 15A72, 33C80 |
| url | https://arxiv.org/abs/2412.03000 |