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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.06633 |
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| _version_ | 1866910881486471168 |
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| author | Liang, Weikang Wang, Suijie Zhao, Chengdong |
| author_facet | Liang, Weikang Wang, Suijie Zhao, Chengdong |
| contents | In this paper, we introduce the $k$-adjoint of a given hyperplane arrangement $\mathcal{A}$ associated with rank-$k$ elements in the intersection lattice $L(\mathcal{A})$, which generalizes the classical adjoint proposed by Bixby and Coullard. The $k$-adjoint of $\mathcal{A}$ induces a decomposition of the Grassmannian, which we call the $\mathcal{A}$-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of $\mathcal{A}$. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the $k$-dimensional restrictions of $\mathcal{A}$. Consequently, we establish the anti-monotonicity property of some combinatorial invariants, such as Whitney numbers of the first kind and the independece numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $k$-Adjoint of Hyperplane Arrangements Liang, Weikang Wang, Suijie Zhao, Chengdong Combinatorics In this paper, we introduce the $k$-adjoint of a given hyperplane arrangement $\mathcal{A}$ associated with rank-$k$ elements in the intersection lattice $L(\mathcal{A})$, which generalizes the classical adjoint proposed by Bixby and Coullard. The $k$-adjoint of $\mathcal{A}$ induces a decomposition of the Grassmannian, which we call the $\mathcal{A}$-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of $\mathcal{A}$. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the $k$-dimensional restrictions of $\mathcal{A}$. Consequently, we establish the anti-monotonicity property of some combinatorial invariants, such as Whitney numbers of the first kind and the independece numbers. |
| title | $k$-Adjoint of Hyperplane Arrangements |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.06633 |