Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections
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| Format: | Preprint |
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2025
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| _version_ | 1866908606609227776 |
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| author | Garnier, Owen Heng, Edmund Licata, Anthony Yacobi, Oded |
| author_facet | Garnier, Owen Heng, Edmund Licata, Anthony Yacobi, Oded |
| contents | Let $Γ$ be a Coxeter diagram and let $J \subseteq Γ$. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of $J$, which is a $J$-relative generalisation of the classical Coxeter arrangement. For $Γ$ of finite-type, we show that its complexified hyperplane complement is a $K(π,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin--Tits group attached to $J$. For general $Γ$ we show that Brink--Howlett's groupoid, which describes normalisers of parabolic subgroups of Coxeter groups, has its universal cover described by the wall-and-chamber structure of the Tits cone intersection. We use this to show that wall crossing sequences satisfy an "atomic Matsumoto relation", generalising a theorem of Ko and answering questions raised by Iyama and Wemyss. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections Garnier, Owen Heng, Edmund Licata, Anthony Yacobi, Oded Group Theory Algebraic Topology Geometric Topology 20F55, 20F36, 55P20 Let $Γ$ be a Coxeter diagram and let $J \subseteq Γ$. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of $J$, which is a $J$-relative generalisation of the classical Coxeter arrangement. For $Γ$ of finite-type, we show that its complexified hyperplane complement is a $K(π,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin--Tits group attached to $J$. For general $Γ$ we show that Brink--Howlett's groupoid, which describes normalisers of parabolic subgroups of Coxeter groups, has its universal cover described by the wall-and-chamber structure of the Tits cone intersection. We use this to show that wall crossing sequences satisfy an "atomic Matsumoto relation", generalising a theorem of Ko and answering questions raised by Iyama and Wemyss. |
| title | Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections |
| topic | Group Theory Algebraic Topology Geometric Topology 20F55, 20F36, 55P20 |
| url | https://arxiv.org/abs/2509.21915 |