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We introduce a theory of jeu de taquin for increasing tableaux, extendingfundamental work of [Sch\"{u}tzenberger '77] for standard Young tableaux. Weapply this to give a new combinatorial rule for the K-theory Schubert calculusof Grassmannians via K-theoretic jeu de taquin, providing an alternative to therules of [Buch '02] and others. This rule naturally generalizes to give aconjectural root-system uniform rule for any minuscule flag variety G/P,extending [Thomas-Yong '06]. We also present analogues of results of Fomin,Haiman, Schensted and Sch\"{u}tzenberger.

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