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

TL;DR

This paper studies the geometric and algebraic properties of open projected Richardson varieties and affine Schubert cells, linking their characteristic classes via pushforward and pullback methods, and explicitly constructs symmetric functions for open positroid varieties.

Contribution

It introduces explicit symmetric functions representing Segre--MacPherson classes of open positroid varieties using pipe dreams for affine permutations.

Findings

Segre--MacPherson classes of open projected Richardson varieties are compared with those of affine Schubert cells.

Explicit symmetric functions for open positroid varieties are constructed in terms of pipe dreams.

The approach connects geometric stratifications with combinatorial models in the affine Grassmannian.

Abstract

The open projected Richardson varieties form a stratification for the partial flag variety $G/P$. We compare the Segre--MacPherson classes of open projected Richardson varieties with those of the corresponding affine Schubert cells by pushing or pulling these classes to the affine Grassmannian. In the case of the Grassmannian $G/P=\operatorname{Gr}_k(\mathbb{C}^n)$, the open projected Richardson varieties are known as open positroid varieties. We obtain symmetric functions that represent the Segre--MacPherson classes of these open positroid varieties, constructed explicitly in terms of pipe dreams for affine permutations.