Monodromy and K-theory of Schubert curves via generalized jeu de taquin

Abstract

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S({\lambda }_{\bullet })$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of ${\mathbb{RP}}^1$, with $\omega$ as the monodromy operator. We provide a local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S({\lambda }_{\bullet })$.

33 pages, 12 figures including 2 color figures; to appear in the Journal of Algebraic Combinatorics

References in corpus (2)

• Genomic Tableaux
• One-dimensional Schubert problems with respect to osculating flags

Cited by in corpus (2)

• Equivariant K-theory of Grassmannians
• The genomic Schur function is fundamental-positive