The direct sum map on grassmannians and jeu de taquin for increasing tableaux

The direct sum map Gr(a,Cⁿ) × Gr(b,C^m) →Gr(a+ b,C^m+n) on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from Buch ["Grothendieck classes of quiver varieties." Duke Mathematical Journal 115, no. 1 (2002): 75-103] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [Thomas and Yong. "A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus." Algebra and Number Theory Journal 3, no. 2 (2009): 121-48] by proving transparent analogues of Schützenberger's ["La Correspondance de Robinson." In Combinatoire et Représentation du Groupe Symétrique (Strasbourg, 1976), edited by D. Foata, 59-113. Lecture Notes in Mathematics 579. Berlin: Springer, 1977] fundamental theorems on well definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients.