Non-representable hyperbolic matroids
The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connect...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6328/pdf |
| _version_ | 1797270273289879552 |
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| author | Nima Amini Petter Branden |
| author_facet | Nima Amini Petter Branden |
| author_sort | Nima Amini |
| collection | DOAJ |
| description | The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture. |
| first_indexed | 2024-04-25T02:01:39Z |
| format | Article |
| id | doaj.art-440d22e29abe4f3cb71143298a6fc6a9 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:39Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-440d22e29abe4f3cb71143298a6fc6a92024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63286328Non-representable hyperbolic matroidsNima Amini0https://orcid.org/0000-0002-2305-9764Petter Branden1Department of MathematicsDepartment of MathematicsThe generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.https://dmtcs.episciences.org/6328/pdf[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Nima Amini Petter Branden Non-representable hyperbolic matroids Discrete Mathematics & Theoretical Computer Science [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Non-representable hyperbolic matroids |
| title_full | Non-representable hyperbolic matroids |
| title_fullStr | Non-representable hyperbolic matroids |
| title_full_unstemmed | Non-representable hyperbolic matroids |
| title_short | Non-representable hyperbolic matroids |
| title_sort | non representable hyperbolic matroids |
| topic | [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6328/pdf |
| work_keys_str_mv | AT nimaamini nonrepresentablehyperbolicmatroids AT petterbranden nonrepresentablehyperbolicmatroids |