Equivariant Schubert calculus and jeu de taquin

We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of Schützenberger’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.