Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914141899325440 |
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| author | Defant, Colin Sherman-Bennett, Melissa Williams, Nathan |
| author_facet | Defant, Colin Sherman-Bennett, Melissa Williams, Nathan |
| contents | For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group Defant, Colin Sherman-Bennett, Melissa Williams, Nathan Combinatorics Group Theory Geometric Topology 20F55, 52B11, 05E45, 20F36 For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group. |
| title | Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group |
| topic | Combinatorics Group Theory Geometric Topology 20F55, 52B11, 05E45, 20F36 |
| url | https://arxiv.org/abs/2509.11497 |