Related papers: Projected Richardson varieties and affine Schubert…

We study algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson…

Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that…

The stratification of the Grassmannian by positroid varieties has been the subject of extensive research. Positroid varieties are in bijection with a number of combinatorial objects, including $k$-Bruhat intervals and bounded affine…

We first describe the tangent space to the affine flag manifold associated to a simple algebraic group over $\mathbb{C}$ at the distinguished point starting from standard definitions. We then construct projective lines in the affine flag…

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal…

We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived…

Let $G$ be a simply connected, almost simple group over an algebraically closed field $\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\phi:T^*G/P\rightarrow…

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the…

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special…

We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of flag manifolds $G/P$, which can be thought of as a replacement for the ``divisor axiom'' in their (torus-equivariant) quantum…

We use the construction of the stable homotopy category by Khan-Ravi to calculate the integral $T$-equivariant $K$-theory spectrum of a flag variety over an affine scheme, where $T$ is a split torus associated to the flag variety. More…

We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$…

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram,…

Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H^*_S(Pet) of the…

One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank…

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point,…

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of $n \times n$ matrices. We give a positive answer to their conjecture…

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,…

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of…

Based on the Basis theorem of Bruhat--Chevalley [C] and the formula for multiplying Schubert classes obtained in [D\QTR{group}{u}] and programed in [DZ$_{\QTR{group}{1}}$], we introduce a new method computing the Chow rings of flag…