K-theoretic Schubert calculus and applications

A central result in algebraic combinatorics is the Littlewood-Richardson rule that governs products in the cohomology of Grassmannians. A major theme of the modern Schubert calculus is to extend this rule and its associated combinatorics to richer cohomology theories. This thesis focuses on K-theoretic Schubert calculus. We prove the first Littlewood-Richardson rule in torus-equivariant K-theory. We thereby deduce the conjectural rule of H. Thomas and A. Yong, as well as a mild correction to the conjectural rule of A. Knutson and R. Vakil. Our rule manifests the positivity established geometrically by D. Anderson, S. Griffeth and E. Miller, and moreover in a stronger 'squarefree' form that resolves an issue raised by A. Knutson. Our work is based on the combinatorics of genomic tableaux, which we introduce, and a generalization of M.-P. Schuetzenberger's jeu de taquin. We further apply genomic tableaux to obtain new rules in non-equivariant K-theory for Grassmannians and maximal orthogonal Grassmannians, as well as to make various conjectures relating to Lagrangian Grassmannians. This is joint work with Alexander Yong. Our theory of genomic tableaux is a semistandard analogue of the increasing tableau theory initiated by H. Thomas and A. Yong. These increasing tableaux carry a natural lift of M.-P. Schuetzenberger's promotion operator. We study the orbit structure of this action, generalizing a result of D. White by establishing an instance of the cyclic sieving phenomenon of V. Reiner, D. Stanton and D. White. In joint work with J. Bloom and D. Saracino, we prove a homomesy conjecture of J. Propp and T. Roby for promotion on standard tableaux, which partially generalizes to increasing tableaux. In joint work with K. Dilks and J. Striker, we relate the action of K-promotion on increasing tableaux to the rowmotion operator on plane partitions, yielding progress on a conjecture of P. Cameron and D. Fon-der-Flaass. Building on this relation between increasing tableaux and plane partitions, we apply the K-theoretic jeu de taquin of H. Thomas and A. Yong to give, in joint work with Z. Hamaker, R. Patrias and N. Williams, the first bijective proof of a 1983 theorem of R. Proctor, namely that that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a trapezoid