A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of Schutzenberger (1977) for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of Buch and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending recent work of Thomas and Yong. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.