Related papers: Schubert decompositions for ind-varieties of gener…
This is a review of results on the structure of the homogeneous ind-varieties $G/P$ of the ind-groups $G=\mathrm{GL}_{\infty}(\mathbb{C})$, $\mathrm{SL}_{\infty}(\mathbb{C})$, $\mathrm{SO}_{\infty}(\mathbb{C})$,…
We generalize the classification of isomorphism classes of Schubert varieties in complete flag varieties G/B to a class of partial flag varieties G/P. In particular, we classify all Schubert varieties in G/P where P is a minimal parabolic…
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…
For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these…
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…
The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of…
Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with…
Let X be the direct product of two Grassmann varieties of k- and l-planes in a finite-dimensional vector space V, and let B be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog to Schubert cells in…
In this paper, we study the subvarieties of a complex flag variety that are invariant under the action of a maximal torus. Using combinatorial techniques derived from matroid theory, we introduce a decomposition of this variety into affine,…
We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants…
This paper focuses on the properties of Schubert cells as quasi-projective subvarieties of a generalized flag variety. More specifically, we investigate the problem of distinguishing between different Schubert cells using vanishing patterns…
We consider the complex ind-group $G=\mathrm{SL}(\infty,\mathbb{C})$ and its real forms $G^0=\mathrm{SU}(\infty,\infty)$, $\mathrm{SU}(p,\infty)$, $\mathrm{SL}(\infty,\mathbb{R})$, $\mathrm{SL}(\infty,\mathbb{H})$. Our main objects of study…
We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert cells defined by such flags meet transversally. This strengthens a result of…
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main…
Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized…
We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in…
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 obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G/B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a…
We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model…
A Schubert variety in the complete flag manifold $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has a dense orbit. We give a combinatorial classification of these Schubert varieties.…