Cylindrical Networks and Total Nonnegativity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913699210461184 |
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| author | Angarone, Robert |
| author_facet | Angarone, Robert |
| contents | We prove that an infinite block-Toeplitz matrix with finite diagonal support is totally nonnegative if and only if it is the weight matrix of a cylindrical network. This generalizes a well-known theorem of Brenti concerning finite totally nonnegative matrices and planar networks; in particular, our work gives an alternative, self-contained proof of the non-square case. Our argument employs Temperley-Lieb immanants, first introduced by Rhoades and Skandera, which are certain elements of Lusztig's dual canonical bases. As an application, we also obtain a new proof of a well-known theorem relating totally nonnegative block-Toeplitz matrices to interlacing polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09385 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cylindrical Networks and Total Nonnegativity Angarone, Robert Combinatorics Classical Analysis and ODEs 15B48 (Primary), 05E99, 15B05, 05C50 (Secondary) We prove that an infinite block-Toeplitz matrix with finite diagonal support is totally nonnegative if and only if it is the weight matrix of a cylindrical network. This generalizes a well-known theorem of Brenti concerning finite totally nonnegative matrices and planar networks; in particular, our work gives an alternative, self-contained proof of the non-square case. Our argument employs Temperley-Lieb immanants, first introduced by Rhoades and Skandera, which are certain elements of Lusztig's dual canonical bases. As an application, we also obtain a new proof of a well-known theorem relating totally nonnegative block-Toeplitz matrices to interlacing polynomials. |
| title | Cylindrical Networks and Total Nonnegativity |
| topic | Combinatorics Classical Analysis and ODEs 15B48 (Primary), 05E99, 15B05, 05C50 (Secondary) |
| url | https://arxiv.org/abs/2312.09385 |