Hyperplane arrangements and Vinberg's $θ$-groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912985974308864 |
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| author | Ambrosio, Filippo Santi, Andrea |
| author_facet | Ambrosio, Filippo Santi, Andrea |
| contents | Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c} \subset \mathfrak{g}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_04284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperplane arrangements and Vinberg's $θ$-groups Ambrosio, Filippo Santi, Andrea Representation Theory Group Theory 17B70 (Primary), 17B40, 20F55 (Secondary) Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c} \subset \mathfrak{g}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$. |
| title | Hyperplane arrangements and Vinberg's $θ$-groups |
| topic | Representation Theory Group Theory 17B70 (Primary), 17B40, 20F55 (Secondary) |
| url | https://arxiv.org/abs/2509.04284 |