Maria Monks Gillespie ; Jake Levinson

Monodromy and Ktheory of Schubert curves via generalized jeu de taquin
dmtcs:6381 
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

https://doi.org/10.46298/dmtcs.6381
Monodromy and Ktheory of Schubert curves via generalized jeu de taquinArticle
Authors: Maria Monks Gillespie ^{1}; Jake Levinson ^{2}
We establish a combinatorial connection between the real geometry and the Ktheory of complex Schubert curves Spλ‚q, which are onedimensional 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 Ktheoretic LittlewoodRichardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the Ktheory and real geometry of Spλ‚q.