Schubert Curves in the Orthogonal Grassmannian.

We develop a combinatorial rule to compute the real geometry of type B Schubert curves S (λ ∙) in the orthogonal Grassmannian O G (n , C 2 n + 1) , which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A [J. Algebraic Combin. 45(1), 191–243 (2017)]. First, using the type B Wroński map studied in [Adv. Math. 224(3), 827–862 (2010)], we show that the real locus of the Schubert curve has a natural covering map to R P 1 , with monodromy operator ω defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute ω without rectifying the skew tableau. The first uses the crystal operators introduced in [Algebr. Comb. 3(3), 693–725 (2020)], while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves. [ABSTRACT FROM AUTHOR]