Characteristic quasi-polynomials of deletions of Shi arrangements of type B and their period collapse
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914308578869248 |
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| author | Higashitani, Akihiro Nakashima, Norihiro |
| author_facet | Higashitani, Akihiro Nakashima, Norihiro |
| contents | Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers. A notable phenomenon in this context is period collapse, where the quasi-polynomial reduces to a polynomial or has a smaller period than the lcm period. In this paper, we compute the characteristic quasi-polynomials of the restriction of the Shi arrangement of type B by one given hyperplane. As a corollary, we completely determine whether period collapse occurs in the characteristic quasi-polynomial of the deletion of the Shi arrangement of type B. This implies the solution for the conjecture posed by Higashitani, Tran and Yoshinaga in this case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20102 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characteristic quasi-polynomials of deletions of Shi arrangements of type B and their period collapse Higashitani, Akihiro Nakashima, Norihiro Combinatorics 32S22 (Primary), 52C35 (Secondary) Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers. A notable phenomenon in this context is period collapse, where the quasi-polynomial reduces to a polynomial or has a smaller period than the lcm period. In this paper, we compute the characteristic quasi-polynomials of the restriction of the Shi arrangement of type B by one given hyperplane. As a corollary, we completely determine whether period collapse occurs in the characteristic quasi-polynomial of the deletion of the Shi arrangement of type B. This implies the solution for the conjecture posed by Higashitani, Tran and Yoshinaga in this case. |
| title | Characteristic quasi-polynomials of deletions of Shi arrangements of type B and their period collapse |
| topic | Combinatorics 32S22 (Primary), 52C35 (Secondary) |
| url | https://arxiv.org/abs/2405.20102 |