Bases of the Intersection Cohomology of Grassmannian Schubert Varieties

TL;DR

This paper connects combinatorial formulas for Grassmannian Kazhdan-Lusztig polynomials with the structure of singular Soergel bimodules, providing explicit bases for intersection cohomology of Schubert varieties.

Contribution

It introduces a novel categorical lifting of Dyck partition formulas to construct bases of Hom spaces in singular Soergel bimodules, extending classical cohomology bases.

Findings

Bases of intersection cohomology parametrized by Dyck partitions

Categorical interpretation of Kazhdan-Lusztig polynomials

Extension of classical Schubert bases to singular settings

Abstract

The parabolic Kazhdan-Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We "lift" this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces between indecomposable objects. In particular, we describe bases of intersection cohomology of Schubert varieties in Grassmannians parametrized by Dyck partitions which extend (after dualizing) the classical Schubert basis of the ordinary cohomology.