Schubert curves in the orthogonal Grassmannian

TL;DR

This paper introduces a combinatorial rule for analyzing the real geometry of type B Schubert curves in the orthogonal Grassmannian, extending known results from type A and providing algorithms for computation.

Contribution

It develops new combinatorial algorithms to compute the monodromy of type B Schubert curves and links these to K-theory, advancing understanding of their geometry.

Findings

Real locus of Schubert curve covers $ ext{RP}^1$ with monodromy $ ext{omega}$

Algorithms using shifted tableau crystal operators and local switches to compute $ ext{omega}$

Switching algorithm enumerates shifted genomic tableaux related to K-theory

Abstract

We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, 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. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $\omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $\omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, 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.

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