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

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 Levinson (One-dimensional Schubert problems with respect to osculating flags, 2016, doi: 10.4153/CJM-2015-061-1), it was shown 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 fast, 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 (Pechenik and Yong in Genomic tableaux, 2016. arXiv:1603.08490), 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 \(S(\lambda _\bullet )\).