Explorations of Matroid Complexes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917528319557632 |
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| author | Bruce, Juliette Bucciarelli, Jacob Zacovic, Bailee |
| author_facet | Bruce, Juliette Bucciarelli, Jacob Zacovic, Bailee |
| contents | Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explorations of Matroid Complexes Bruce, Juliette Bucciarelli, Jacob Zacovic, Bailee Combinatorics Commutative Algebra Algebraic Geometry Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids. |
| title | Explorations of Matroid Complexes |
| topic | Combinatorics Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2605.24695 |