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We prove Pieri formulas for the multiplication with special Schubert classes in the  K -theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood–Richardson rule of Thomas and Yong. Recent work of Clifford, Thomas, and Yong has shown that the full Littlewood–Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the  K -theoretic Pieri coefficients both as integers determined by positive recursive identities and as the number of certain tableaux. The proof is based on a computation of the sheaf Euler characteristic of triple intersections of Schubert varieties, where at least one Schubert variety is special.

Pieri rules for the K -theory of cominuscule Grassmannians