The external activity complex of a pair of matroids
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910891245568000 |
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| author | Berget, Andrew Fink, Alex |
| author_facet | Berget, Andrew Fink, Alex |
| contents | We introduce the Schubert variety of a pair of linear subspaces in $\mathbf{C}^n$ and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when one of the linear spaces is one-dimensional. We prove that our external activity complex is Cohen-Macaulay and deduce a formula for its $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of matroids. As a consequence, we deduce a non-negative formula for the matroid invariant $ω(M)$ of Fink, Shaw, and Speyer in terms of certain homology groups of links within an external activity complex, proving the 2005 tropical $f$-vector conjecture of Speyer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The external activity complex of a pair of matroids Berget, Andrew Fink, Alex Combinatorics Commutative Algebra Algebraic Geometry 05B35 (Primary) 13D02, 13F55, 14T15 (Secondary) We introduce the Schubert variety of a pair of linear subspaces in $\mathbf{C}^n$ and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when one of the linear spaces is one-dimensional. We prove that our external activity complex is Cohen-Macaulay and deduce a formula for its $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of matroids. As a consequence, we deduce a non-negative formula for the matroid invariant $ω(M)$ of Fink, Shaw, and Speyer in terms of certain homology groups of links within an external activity complex, proving the 2005 tropical $f$-vector conjecture of Speyer. |
| title | The external activity complex of a pair of matroids |
| topic | Combinatorics Commutative Algebra Algebraic Geometry 05B35 (Primary) 13D02, 13F55, 14T15 (Secondary) |
| url | https://arxiv.org/abs/2412.11759 |