EQUIVARIANT SCHUBERT CALCULUS AND JEU DE TAQUIN

We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of Schiitzenberger's theory of jeu de taquin. These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we present new (semi)standard tableaux to study factorial Schur polynomials, after Biedenharn-Louck, Macdonald, Goulden-Greene, and others. Consequently, we obtain new combinatorial rules for the Schubert structure coefficients, complementing work of Molev-Sagan, Knutson-Tao, Molev, and Kreiman. We also describe a conjectural generalization of one of our rules to the equivariant K-theory of Grassmannians, extending our previous work on non-equivariant K-theory. This conjecture concretely realizes the positivity known to exist by a result of Anderson-Griffeth-Miller. It provides an alternative to the conjectural rule of Knutson-Vakil.