On Bott--Samelson rings for Coxeter groups

TL;DR

This paper provides an explicit combinatorial presentation of the cohomology ring of Bott--Samelson varieties for Coxeter groups, establishing its algebraic properties and geometric features such as the Kähler package.

Contribution

It introduces the Bott--Samelson ring for arbitrary Coxeter systems with a quadratic complete intersection structure and proves it satisfies key geometric properties.

Findings

The ring is a split quadratic complete intersection algebra.

It is a Koszul algebra with a quadratic Gr{"o}bner basis.

It satisfies the K{"a}hler package, including Poincaré duality, hard Lefschetz, and Hodge--Riemann relations.

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.