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arXiv · 2408.10155

On Bott--Samelson rings for Coxeter groups

Tao Gui·Lin Sun·Shihao Wang·Haoyu Zhu
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Abstract

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{\"o}bner basis. Furthermore, we prove that it satisfies the whole K\"ahler package, including the Poincar\'e duality, the hard Lefschetz theorem, and the Hodge--Riemann bilinear relations.

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Tao Gui, Lin Sun, Shihao Wang, Haoyu Zhu. 2024-08-19. On Bott--Samelson rings for Coxeter groups. https://arxiv.org/abs/2408.10155

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