Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
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| Format: | Preprint |
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2022
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| _version_ | 1866916075649630208 |
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| author | Douvropoulos, Theo Lewis, Joel Brewster Morales, Alejandro H. |
| author_facet | Douvropoulos, Theo Lewis, Joel Brewster Morales, Alejandro H. |
| contents | We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of reduced reflection factorizations of $g$ and $\operatorname{RGS}(W,g)$ of the relative generating sets of $g$. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size $\#\operatorname{Red}_W(g)$ with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-$0$ Hurwitz numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_00066 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements Douvropoulos, Theo Lewis, Joel Brewster Morales, Alejandro H. Combinatorics Differential Geometry Group Theory 05Axx, 20F55, 53D45 We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of reduced reflection factorizations of $g$ and $\operatorname{RGS}(W,g)$ of the relative generating sets of $g$. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size $\#\operatorname{Red}_W(g)$ with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-$0$ Hurwitz numbers. |
| title | Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements |
| topic | Combinatorics Differential Geometry Group Theory 05Axx, 20F55, 53D45 |
| url | https://arxiv.org/abs/2209.00066 |