Related papers: Criteria for rational smoothness of some symmetric…

We characterize the O_{2n} orbits in the flag variety for GL_{2n} with rationally smooth closure via a graph-theoretic criterion. We also give a necessary pattern avoidance criterion for rational smoothness and conjecture its sufficiency.

We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the…

With the help of Lusztig's canonical basis, we study local intersection cohomology of the Zariski closures of orbits of representations of a quiver of type A, D or E. In particular, we characterize the rationally smooth orbits and prove…

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…

Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a…

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we…

We give a necessary and sufficient condition in terms of pattern avoidance for the conjugates of the bottom vertex in the Bruhat graph attached to an O_n orbit O in the flag variety for GL_n to have degree equal to the rank of this graph as…

We give a pattern avoidance criterion to classify the orbits of Sp(p,C) x Sp(q,C) (resp. GL(n,C)) on the flag variety of type C_{p+q} (resp. D_n) with rationally smooth closure. We show that all such orbit closures fiber (with smooth fiber)…

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds $G/B$ in terms of patterns in root systems. We generalize Lakshmibai-Sandhya's well-known result that says that a Schubert…

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure,…

We give conceptual and combinatorial criteria for the normality and Cohen--Macaulayness of unions of Ekedahl--Oort strata in the special fiber of abelian type Shimura varieties. For unions of two strata, one of the two having codimension…

Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the…

Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert…

This paper is a continuation of arXiv:1201.1102. We investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on the simple Lie algebra of type $E_7$. The methods for…

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these…

Let $k$ be a field with a nontrivial discrete valuation which is complete and has perfect residue field. Let $G$ be the group of $k$-rational points of a reductive, linear algebraic group $\textbf{G}$ equipped with an involution $\theta$…

Let $X$ denote an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$. Let $B$ denote a Borel subgroup of $G$ and let $Z$ denote a $B \times B$-orbit closure in $X$. When the characteristic of $k$…

In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for…

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…

We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of…