Related papers: A remark on Schubert cells and duality of orbits o…

We examine the orbits of the (complex) symplectic group, $Sp_n$, on the flag manifold, $\mathscr{F}\ell(\mathbb{C}^{2n})$, in a very concrete way. We use two approaches: we Gr\"obner degenerate the orbits to unions of Schubert varieties…

This paper is concerned with the study of spaces of naturally defined cycles associated to SL(n,R)-flag domains. These are compact complex submanifolds in open orbits of real semisimple Lie groups in flag domains of their complexification.…

We give positive descriptions for certain Schubert structure constants $c_{u,v}^w$ for the full flag variety in Lie types $C$ and $D$. This is accomplished by first observing that a number of the $K=GL(n,\C)$-orbit closures on these flag…

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of…

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity will be equal to the Akhiezer-Gindikin domain D if S is of…

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of…

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…

It is shown that the cycle space of an arbitrary orbit of a non-Hermitian real form G in a flag manifold $Z=G^\mathbb C/Q$ of its complexification is naturally equivalent to a certain universal domain which depends only on G. This makes use…

Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. Certain group theoretically defined compact complex submanifolds, which are regarded as cycles, are of basic importance for their…

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…

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 $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic…

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…

Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the…

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…

We study the local and global intersection cohomology of the intersection of two Schubert varieties in a flag manifold. In this version some new references are added.

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…

We prove that Schubert and Richardson varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under…

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…

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…