Schubert geometry and special polynomials

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For the flag variety of classical Lie groups, we introduced special polynomials representing the Schubert classes in torus equivariant cohomology ring. We established some fundamental properties for these polynomials. In order to extend this result to torus equivariant K-theory, we introduced special family of polynomials (K-theoretic Q-and P-functions) representing the Schubert classes in the torus equivariant K-theory of classical Grassmannian varieties. We also developed combinatorics of these polynomials. In particular, we discussed a relationship to excited Young diagrams and shifted set-valued tableaux. We introduced a Robinson-Schensted type algorithm for these tableaux. As an application of this algorithm, we proved Pieri type formulas for the K-theoretic Q-functions.