We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: • Pieri rules for the Schubert bases of H∗(Gr) and H∗(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. • A new combinatorial definition for k-Schur functions, which represent the Schubert ba...

In this talk we present some recent results related to Schubert and Grothendieck polynomials. These polynomials represent Schubert classes, which form the natural bases of the cohomology and K-theory of the complex flag variety. We present background information on several combinatorial constructions of Schubert and Grothendieck polynomials. Then we present the solution to a conjecture concerni...

In analogy with transition equations for type Schubert polynomials given by Lascou and Schutzenberger s 14 , we give recursive formulas for computing representatives of the Schubert classes for the isotropic ag manifolds. These representatives are e actly the Schubert polynomials found in 2 . This new approach to nding Schubert polynomials is very closely related to the geometry of the ag manif...

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type B (respectively, type C) Schubert polynomial by the Schur P -polynomial pm (respectively, the Schur Q-polynomial qm). Geometric constructions and interme...

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with posit...

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here, we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

I construct a correspondence between the Schubert cycles on the variety of complete flags in Cn and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SLn(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein–Gelfand– Gelfand [2] using Demazure modules. The correspondence between ...