Related papers: Chern-Schwartz-MacPherson classes for Schubert cel…
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to…
The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold $X=G/B$ is an element in the equivariant K theory ring of $X$ to which one adjoins a formal parameter $y$. In this paper we prove several `folklore…
The classical Schubert cells on a flag manifold G/H give a cell decomposition for G/H whose Kronecker duals (known as Schubert classes) form an additive base for the integral cohomology H^{\ast}(G/H). We present a formula that expresses…
We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving…
We describe the integral cohomology rings of the flag manifolds of types B_n, D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an…
The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of…
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies…
We study the Demazure-Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomology), respectively their…
Let $G$ be a compact connected Lie group and $T$ be its maximal torus. The homogeneous space $G/T$ is called the (complete) flag manifold. One of the main goals of the {\em equivariant Schubert calculus} is to study the $T$-equivariant…
We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold $G/P$ by the class of any line bundle $\mathcal{L}_\lambda$. Our formula is given in terms of the $\lambda$-chains of Lenart…
Motivic Chern classes are elements in the K-theory of an algebraic variety $X$, depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic…
A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson…
Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{\Theta }$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the…
Consider the natural torus action on a partial flag manifold $Fl$. Let $\Omega_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of…
The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class.…
Based on the multiplicative rule of Schubert classes obtained in [Du3], we present an algorithm computing the product of two arbitrary Schubert classes. As a result, the algorithm gives also a method to compute the integral cohomology ring…
Let G be a compact connected Lie group and H, the centralizer of a one-parameter subgroup in G. Combining the ideas of Bott-Samelson resulotions of Schubert varieties and the enumerative formula on a twisted products of 2-spheres obatained…
For a flag manifold $M=G/B$ with the canonical torus action, the $T-$equivariant cohomology is generated by equivariant Schubert classes, with one class $\tau_u$ for every element $u$ of the Weyl group $W$. These classes are determined by…
In this paper we study the T-equivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the Bott-Samelson classes. After setup…
We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}\Sigma$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to…