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

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 them to the affine Grassmannian. In the Grassmannian case, the open projected Richardson varieties are well known as open positroid varieties. We obtain symmetric functions that represent the Segre–MacPherson classes of open positroid varieties, constructed explicitly in terms of pipe dreams for affine permutations.