Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866914948151508992 |
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| author | Douvropoulos, Theo |
| author_facet | Douvropoulos, Theo |
| contents | In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$ associated to a well-generated complex reflection group $W$. Chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of $NC(W)$ and the fibers of a finite quasi-homogeneous morphism, the $LL$ map.
We consider a variant of the $LL$ map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of $NC(W)$. In particular, we extend work of Bessis and Ripoll and enumerate the so-called "primitive factorizations" of the Coxeter element $c$. That is, length additive factorizations of the form $c=w\cdot t_1\cdots t_k$, where $w$ belongs to a given conjugacy class and the $t_i$'s are reflections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_10395 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element Douvropoulos, Theo Combinatorics Algebraic Geometry 05A99, 20F36, 20F55 In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$ associated to a well-generated complex reflection group $W$. Chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of $NC(W)$ and the fibers of a finite quasi-homogeneous morphism, the $LL$ map. We consider a variant of the $LL$ map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of $NC(W)$. In particular, we extend work of Bessis and Ripoll and enumerate the so-called "primitive factorizations" of the Coxeter element $c$. That is, length additive factorizations of the form $c=w\cdot t_1\cdots t_k$, where $w$ belongs to a given conjugacy class and the $t_i$'s are reflections. |
| title | Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element |
| topic | Combinatorics Algebraic Geometry 05A99, 20F36, 20F55 |
| url | https://arxiv.org/abs/1808.10395 |