Genomic tableaux and combinatorial K-theory

We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant K-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller’11]. We thereby deduce an earlier conjecture of [Thomas-Yong’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) K-theory of Grassmannians. From this perspective, we also obtain a new rule for K-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians.