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

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 $G/P={\mathrm{Gr}}_k({\mathbb{C}}^n)$, 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.

arXiv comment27 pages, some typos are corrected

References in corpus (10)

• Total positivity, Grassmannians, and networks
• Equivariant Chern classes of singular algebraic varieties with group actions
• Back stable Schubert calculus
• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• $k$-Schur functions and affine Schubert calculus
• Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds
• Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
• Hook formulae from Segre-MacPherson classes
• Pieri and Murnaghan--Nakayama type Rules for Chern classes of Schubert Cells
• Richardson varieties, projected Richardson varieties and positroid varieties