Exploring the Varchenko Determinant of Partial Cubes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913849677971456 |
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| author | Hochstättler, Winfried Keip, Sophia Yazici, Birol |
| author_facet | Hochstättler, Winfried Keip, Sophia Yazici, Birol |
| contents | The Varchenko matrix is known to have a well-structured determinant for complexes of oriented matroids (COMs). COMs can be characterized as partial cubes that do not have certain forbidden pc-minors. In this work, we generalize the Varchenko matrix and its determinant to partial cubes. We identify examples of partial cubes whose Varchenko determinants lack a clean factorization, as well as those that exhibit such a structure. These findings open the door for further research into the properties and potential characterizations of partial cubes with well-behaved Varchenko determinants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exploring the Varchenko Determinant of Partial Cubes Hochstättler, Winfried Keip, Sophia Yazici, Birol Combinatorics 52C40, 52C35, 05B35, 05E14 The Varchenko matrix is known to have a well-structured determinant for complexes of oriented matroids (COMs). COMs can be characterized as partial cubes that do not have certain forbidden pc-minors. In this work, we generalize the Varchenko matrix and its determinant to partial cubes. We identify examples of partial cubes whose Varchenko determinants lack a clean factorization, as well as those that exhibit such a structure. These findings open the door for further research into the properties and potential characterizations of partial cubes with well-behaved Varchenko determinants. |
| title | Exploring the Varchenko Determinant of Partial Cubes |
| topic | Combinatorics 52C40, 52C35, 05B35, 05E14 |
| url | https://arxiv.org/abs/2505.14800 |