Ideal Decomposition of Hyperplane Arrangements
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915765189345280 |
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| author | Chen, Yanru Liang, Weikang Wang, Suijie Zhao, Chengdong |
| author_facet | Chen, Yanru Liang, Weikang Wang, Suijie Zhao, Chengdong |
| contents | Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $χ_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to propose combinatorial structures for the factorization of $χ_{\mathcal{A}}(t)$. To this end, we introduce the notion of an ideal decomposition of $L(\mathcal{A})$ and use the Möbius algebra as a key tool to derive such a factorization. This concept provides a unified and substantial generalization of both the modular elements proposed by Stanley (1971) and the nice partitions proposed by Terao (1992). We also define modular ideals of $L(\mathcal{A})$, which yield a tensor decomposition of the Orlik-Solomon algebra of $\mathcal{A}$. We further show that every modular ideal can be realized as the intersection poset of some hyperplane arrangement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ideal Decomposition of Hyperplane Arrangements Chen, Yanru Liang, Weikang Wang, Suijie Zhao, Chengdong Combinatorics Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $χ_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to propose combinatorial structures for the factorization of $χ_{\mathcal{A}}(t)$. To this end, we introduce the notion of an ideal decomposition of $L(\mathcal{A})$ and use the Möbius algebra as a key tool to derive such a factorization. This concept provides a unified and substantial generalization of both the modular elements proposed by Stanley (1971) and the nice partitions proposed by Terao (1992). We also define modular ideals of $L(\mathcal{A})$, which yield a tensor decomposition of the Orlik-Solomon algebra of $\mathcal{A}$. We further show that every modular ideal can be realized as the intersection poset of some hyperplane arrangement. |
| title | Ideal Decomposition of Hyperplane Arrangements |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.12226 |