Related papers: Restricting Schubert classes
Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively…
Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $\lambda=(\lambda_1,...,\lambda_k)$ with $\lambda_1\leq n$…
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of Ghorpade-Raghavan, which gives an…
Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in…
We classify rigid Schubert classes in orthogonal Grassmannians. More generally, given a representative $X$ of a Schubert class in an orthogonal Grassmannian, we give combinatorial conditions which guarantee that every linear space…
The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by…
In an earlier work, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction…
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai,…
Let $H$ be a separable real Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We…
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on…
Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class…
We give a new Littlewood-Richardson rule for the Schubert structure coefficients of isotropic Grassmannians, equivalently for the multiplication of $P$-Schur functions. Serrano (2010) previously gave a formula in terms of classes in his…
In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Pl$\ddot{u}$cker coordinates,…
We study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a sum of multi Schur-Pfaffians, whose entries…
Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure…
The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space…
In this paper, we study the multi-rigidity problem in rational homogeneous spaces. A Schubert class is called multi-rigid if every multiple of it can only be represented by a union of Schubert varieties. We prove the multi-rigidity of…
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order…
Let $L$ be a Levi subgroup of $GL_N$ which acts by left multiplication on a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. We say that $X(w)$ is a spherical Schubert variety if $X(w)$ is a spherical variety for the action of $L$. In…
We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is…