Related papers: Schubert curves in the orthogonal Grassmannian

Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively…

Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number 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…

This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the…

The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. This sparked…

Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to…

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric…

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n+2}$. In this…

Let $T$ be a torus acting on $\CC^n$ in such a way that, for all $1\leq k\leq n$, the induced action on the grassmannian $G(k,n)$ has only isolated fixed points. This paper proposes a natural, elementary, explicit description of the…

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of…

A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are…

Points in the intersection of Schubert varieties are counted by various combinatorial objects, for example standard tableaux. This paper consider the problem of producing a natural labelling of intersection points by these combinatorial…

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…

For any complex reductive connected Lie group G, many of the structure constants of the ordinary cohomology ring H^*(G/B; Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the ``root…

We establish a congruence modulo four in the real Schubert calculus on the Grassmannian of m-planes in 2m-space. This congruence holds for fibers of the Wronski map and a generalization to what we call symmetric Schubert problems. This…

Based on Thomas and Yong's K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the…

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space…

Let V be a 2n-dimensional complex symplectic space. Let G' be the Lagrangian Grassmannian of maximal isotropic subspaces of V embedded via the inclusion i into the Grassmannian G of all n-dimensional subspaces of V. We discuss the…

Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a…

The skew Schubert polynomials are those which are indexed by skew elements of the Weyl group, in the sense of arXiv:0812.0639. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where…