Motivic Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells

TL;DR

This paper studies the motivic Chern classes of open projected Richardson varieties and affine Schubert cells, establishing recursive relations and combinatorial formulas, and relating these classes to Kazhdan--Lusztig polynomials.

Contribution

It introduces a recursive method to compare motivic Chern classes of these varieties and provides explicit combinatorial formulas in the Grassmannian case.

Findings

Recursive relations via Demazure--Lusztig operators

Connection between SMC classes and Kazhdan--Lusztig R-polynomials

Explicit combinatorial formulas for Grassmannian cases

Abstract

The open projected Richardson varieties are images of the open Richardson varieties of the complete flag variety under the canonical projection to the partial flag variety. Our main result compares the Segre motivic Chern (SMC) classes of the open projected Richardson varieties with those of the affine Schubert cells by pushing or pulling these classes to the affine Grassmannian. The main method is the recursive relation determined by the Demazure--Lusztig operators. As another application of this recursive relation, we relate the localization of the SMC classes to the twisted Kazhdan--Lusztig R-polynomials. In the case of Grassmannians, the open projected Richardson varieties are known as the open positroid varieties. We give a combinatorial formula for the SMC classes of these varieties.