The hyperbolic cover of an elliptic Weyl group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918218474455040 |
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| author | Baumeister, Barbara Wegener, Patrick |
| author_facet | Baumeister, Barbara Wegener, Patrick |
| contents | In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For $\tilde{c}$ a Coxeter transformation in $\tilde{W}$ we can conclude the Hurwitz transitivity of the braid group action on the set of reduced reflection factorizations of $\tilde{c}$ from the Hurwitz transitivity in extended Coxeter systems of star type. This then enables us to establish for a weighted projective line $\mathbb{X}$ of tubular type an order preserving bijection between the poset of thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and the poset $[\mathrm{id}, \tilde{c}]$ ordered by the absolute order. In an Appendix, we study the hyperbolic cover of a Coxeter system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_06401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The hyperbolic cover of an elliptic Weyl group Baumeister, Barbara Wegener, Patrick Group Theory Representation Theory In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For $\tilde{c}$ a Coxeter transformation in $\tilde{W}$ we can conclude the Hurwitz transitivity of the braid group action on the set of reduced reflection factorizations of $\tilde{c}$ from the Hurwitz transitivity in extended Coxeter systems of star type. This then enables us to establish for a weighted projective line $\mathbb{X}$ of tubular type an order preserving bijection between the poset of thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and the poset $[\mathrm{id}, \tilde{c}]$ ordered by the absolute order. In an Appendix, we study the hyperbolic cover of a Coxeter system. |
| title | The hyperbolic cover of an elliptic Weyl group |
| topic | Group Theory Representation Theory |
| url | https://arxiv.org/abs/2411.06401 |