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

Maria Monks Gillespie (1) , Jake Levinson (2)

1 UC Berkeley - University of California [Berkeley]
2 University of Michigan [Ann Arbor]

We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, 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 ω 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 RP1, with ω as the monodromy operator. We provide a fast, local algorithm for computing ω 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. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

Domaines: Combinatoire [math.CO]

Maria Monks Gillespie, Jake Levinson. Monodromy and K-theory of Schubert curves via generalized jeu de taquin. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6381⟩. ⟨hal-02173066⟩