On Bott--Samelson rings for Coxeter groups

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.