Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case
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arXiv
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| Format: | Preprint |
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2018
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| author | Baumeister, Barbara Wegener, Patrick Yahiatene, Sophiane |
| author_facet | Baumeister, Barbara Wegener, Patrick Yahiatene, Sophiane |
| contents | We start the systematic study of extended Weyl groups, and continue the combinatorial description of thick subcategories in hereditary categories started by Ingalls-Thomas, Igusa-Schiffler-Thomas and Krause. We show that for a weighted projective line $\mathbb{X}$ there exists an order preserving bijection between the thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and a subposet of the interval poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced extended root system if the Hurwitz action is transitive on the reduced reflection factorizations of $c$ that generate the Weyl group. By using combinatorial and group theoretical tools we show that this assumption on the transitivity of the Hurwitz action is fulfilled for a weighted projective line $\mathbb{X}$ of tubular type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_05083 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case Baumeister, Barbara Wegener, Patrick Yahiatene, Sophiane Representation Theory Combinatorics Group Theory We start the systematic study of extended Weyl groups, and continue the combinatorial description of thick subcategories in hereditary categories started by Ingalls-Thomas, Igusa-Schiffler-Thomas and Krause. We show that for a weighted projective line $\mathbb{X}$ there exists an order preserving bijection between the thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and a subposet of the interval poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced extended root system if the Hurwitz action is transitive on the reduced reflection factorizations of $c$ that generate the Weyl group. By using combinatorial and group theoretical tools we show that this assumption on the transitivity of the Hurwitz action is fulfilled for a weighted projective line $\mathbb{X}$ of tubular type. |
| title | Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case |
| topic | Representation Theory Combinatorics Group Theory |
| url | https://arxiv.org/abs/1808.05083 |