Simplicial generation of Chow rings of matroids
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866917957311922176 |
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| author | Backman, Spencer Eur, Christopher Simpson, Connor |
| author_facet | Backman, Spencer Eur, Christopher Simpson, Connor |
| contents | We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of [AHK18]. Our work avoids the use of "flips," the key technical tool employed in [AHK18]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1905_07114 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Simplicial generation of Chow rings of matroids Backman, Spencer Eur, Christopher Simpson, Connor Combinatorics Commutative Algebra Algebraic Geometry 05B35, 52B40, 14T05, 14C17, 14M25 We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of [AHK18]. Our work avoids the use of "flips," the key technical tool employed in [AHK18]. |
| title | Simplicial generation of Chow rings of matroids |
| topic | Combinatorics Commutative Algebra Algebraic Geometry 05B35, 52B40, 14T05, 14C17, 14M25 |
| url | https://arxiv.org/abs/1905.07114 |