The N\'eron-Severi Lie Algebra of a Soergel Module

TL;DR

This paper introduces the Néron-Severi Lie algebra for Soergel modules, determines it for many Schubert varieties, and uses it to give a simple proof of a classical smoothness criterion based on Betti numbers.

Contribution

It defines the Néron-Severi Lie algebra for Soergel modules and applies it to characterize the rational smoothness of Schubert varieties.

Findings

Determined the Néron-Severi Lie algebra for a large class of Schubert varieties.

Connected the algebraic structure to Poincaré duality and smoothness criteria.

Provided a new, simplified proof of a classical smoothness characterization.

Abstract

We introduce the N\'eron-Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the N\'eron-Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincar\'e duality.