On Bott--Samelson rings for Coxeter groups
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910684687630336 |
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| author | Gui, Tao Sun, Lin Wang, Shihao Zhu, Haoyu |
| author_facet | Gui, Tao Sun, Lin Wang, Shihao Zhu, Haoyu |
| contents | We study the cohomology ring of the Bott--Samelson variety. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a purely combinatorial invariant. We use the presentation to introduce the Bott--Samelson ring associated with a word in arbitrary Coxeter system by generators and relations. In general, it is a split quadratic complete intersection algebra with a triangular pattern of relations. By a result of Tate, it follows that it is a Koszul algebra and we provide a quadratic (reduced) Gr{ö}bner basis. Furthermore, we prove that it satisfies the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge--Riemann bilinear relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Bott--Samelson rings for Coxeter groups Gui, Tao Sun, Lin Wang, Shihao Zhu, Haoyu Rings and Algebras Combinatorics 20F55, 14D07, 16S37, 14M15 We study the cohomology ring of the Bott--Samelson variety. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a purely combinatorial invariant. We use the presentation to introduce the Bott--Samelson ring associated with a word in arbitrary Coxeter system by generators and relations. In general, it is a split quadratic complete intersection algebra with a triangular pattern of relations. By a result of Tate, it follows that it is a Koszul algebra and we provide a quadratic (reduced) Gr{ö}bner basis. Furthermore, we prove that it satisfies the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge--Riemann bilinear relations. |
| title | On Bott--Samelson rings for Coxeter groups |
| topic | Rings and Algebras Combinatorics 20F55, 14D07, 16S37, 14M15 |
| url | https://arxiv.org/abs/2408.10155 |