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The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians inducesa K-theory pullback that defines the splitting coefficients. We geometricallyexplain an identity from [Buch '02] between the splitting coefficients and theSchubert structure constants for products of Schubert structure sheaves. Thisis related to the topic of product and splitting coefficients for Schubertboundary ideal sheaves. Our main results extend jeu de taquin for increasingtableaux [Thomas-Yong '09] by proving transparent analogues of[Sch\"{u}tzenberger '77]'s fundamental theorems on well-definedness ofrectification. We then establish that jeu de taquin gives rules for each ofthese four kinds of coefficients.

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