The virtual Euler characteristic for binary matroids
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912873850077184 |
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| author | Brandt, Madeline Bruce, Juliette Corey, Daniel |
| author_facet | Brandt, Madeline Bruce, Juliette Corey, Daniel |
| contents | Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of matroids of rank $r$ realizable over $\mathbb{F}_2$ (i.e., binary matroids). We prove this formula by relating the virtual Euler characteristic for binary matroids to the point counts of certain subsets of Grassmanians over finite fields. We conclude by providing several follow-up questions in relation to matroids realizable over other finite prime fields, matroid homology, and beta invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10108 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The virtual Euler characteristic for binary matroids Brandt, Madeline Bruce, Juliette Corey, Daniel Combinatorics Algebraic Geometry 05B35 (primary), 05A10, 14M15, 15B33 (secondary) Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of matroids of rank $r$ realizable over $\mathbb{F}_2$ (i.e., binary matroids). We prove this formula by relating the virtual Euler characteristic for binary matroids to the point counts of certain subsets of Grassmanians over finite fields. We conclude by providing several follow-up questions in relation to matroids realizable over other finite prime fields, matroid homology, and beta invariants. |
| title | The virtual Euler characteristic for binary matroids |
| topic | Combinatorics Algebraic Geometry 05B35 (primary), 05A10, 14M15, 15B33 (secondary) |
| url | https://arxiv.org/abs/2301.10108 |