Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913283415474176 |
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| author | Abe, Takuro Tran, Tan Nhat |
| author_facet | Abe, Takuro Tran, Tan Nhat |
| contents | Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this paper, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This is a completion of the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan Abe, Takuro Tran, Tan Nhat Combinatorics Primary: 52C35, Secondary: 17B22 Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this paper, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This is a completion of the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems. |
| title | Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan |
| topic | Combinatorics Primary: 52C35, Secondary: 17B22 |
| url | https://arxiv.org/abs/2403.17274 |