Related papers: Levi-spherical Schubert varieties

We obtain a geometric construction of a ``standard monomial basis'' for the homogeneous coordinate ring associated with any ample line bundle on any flag variety. This basis is compatible with Schubert varieties, opposite Schubert…

Let L be k((\epsilon)), where k is an algebraic closure of a finite field with q elements and \epsilon is an indeterminate, and let \sigma be the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of…

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…

Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so…

We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.

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…

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg…

Let $G$ be a connected, complex reductive Lie group and $G/H$ a spherical homogenous space. Let $(X,L)$ be a polarized $G$-variety which is a spherical embedding of $G/H$. In this paper we classify $G$-equivariant normal $\mathbb R$-test…

We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincar\'e polynomial, (2) characterize when these varieties are irreducible,…

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple…

Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module…

Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.

Let G be a connected complex semisimple Lie group. Webster and Yakimov have constructed partitions of the double flag variety G/B x G/B_, where (B, B_) is a pair of opposite Borel subgroups of G, generalizing the Deodhar decompositions of…

It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recent preprint,…

The versal deformation ring R(G,V) of a mod p representation V of a profinite group G encodes all isomorphism classes of lifts of V to representations of G over complete local commutative Noetherian rings. We introduce a new technique for…

In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally…

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…

We prove an explicit degree formula for certain unitary Deligne-Lusztig varieties. Combining with an alternative degree formula in terms of Schubert calculus, we deduce several algebraic combinatorial identities which may be of independent…

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new…