Totally positive skew-symmetric matrices
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929644625723392 |
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| author | Boretsky, Jonathan Cortes, Veronica Calvo Maazouz, Yassine El |
| author_facet | Boretsky, Jonathan Cortes, Veronica Calvo Maazouz, Yassine El |
| contents | A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian $\mathrm{OGr}(n,2n)$. Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Totally positive skew-symmetric matrices Boretsky, Jonathan Cortes, Veronica Calvo Maazouz, Yassine El Combinatorics Algebraic Geometry 14M15, 15B48, 05E14 A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian $\mathrm{OGr}(n,2n)$. Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid. |
| title | Totally positive skew-symmetric matrices |
| topic | Combinatorics Algebraic Geometry 14M15, 15B48, 05E14 |
| url | https://arxiv.org/abs/2412.17233 |