Combinatorics of cone types in Coxeter groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915978383720448 |
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| author | Yau, Yeeka |
| author_facet | Yau, Yeeka |
| contents | In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $β$ in its inversion set $Φ(x)$, the set of elements $y \in W$ satisfying $Φ(x) \cap Φ(y) = \{ β\}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorics of cone types in Coxeter groups Yau, Yeeka Group Theory Combinatorics 20F55, 20F10 In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $β$ in its inversion set $Φ(x)$, the set of elements $y \in W$ satisfying $Φ(x) \cap Φ(y) = \{ β\}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type. |
| title | Combinatorics of cone types in Coxeter groups |
| topic | Group Theory Combinatorics 20F55, 20F10 |
| url | https://arxiv.org/abs/2410.22599 |