Related papers: Schubert decompositions for ind-varieties of gener…

This paper generalizes the results of the paper \cite{mi3} to the case of the general $\mathfrak{sl}_2$ Schubert varieties. We study the homomorphisms between different Schubert varieties, describe their geometry and the group of the line…

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…

We give a closed-form formula for the Hilbert function of the tangent cone at the identity of a Schubert variety X in the Grassmannian in both group theoretic and combinatorial terms. We also give a formula for the multiplicity of X at the…

These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag…

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a…

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a…

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a…

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in…

Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P and use this to give a more…

We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks.…

Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related…

In this article, we use the Bruhat and Schubert cells to calculate the cellular homology of the maximal compact subgroup $K$ of a connected semisimple Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal compact…

For each $A\in\N^n$ we define a Schubert variety $\sh_A$ as a closure of the $\Slt(\C[t])$-orbit in the projectivization of the fusion product $M^A$. We clarify the connection of the geometry of the Schubert varieties with an algebraic…

We provide a method for gluing (small) resolutions of singularities of Schubert varieties \(X_w\). An explicit isomorphism of \(X_w\) with an (iterated) bundle is constructed when \(w\) has an (iterated) BP decomposition. Combined with the…

In this paper, we study the homogeneous components of the Chern--Schwartz--MacPherson (CSM) classes of Schubert cells. We prove that, under suitable conditions, each such component is represented by an irreducible subvariety. In particular,…

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…

Let G be a complex semi-simple Lie group and let P,Q be a pair of parabolic subgroups of G such that Q contains P. Consider the flag varieties G/P, G/Q and Q/P. We show that certain structure constants in H^*(G/P) with respect to the…

Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We…

Let $G/P$ be a complex cominuscule flag manifold of type $E_6,E_7$. We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in $G/P$ is irreducible. The proof utilizes an earlier algorithm by…

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the…