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# One-dimensional Schubert Problems with Respect to Osculating Flags

Published:2016-07-19
Printed: Feb 2017
• Jake Levinson,
Mathematics Department , University of Michigan, Ann Arbor, MI
 Format: LaTeX MathJax PDF

## Abstract

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $\overline{\mathcal{M}_{0,r}}$, allowing the points to collide. These give rise to smooth covers of $\overline{\mathcal{M}_{0,r}} (\mathbb{R})$, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{\mathcal{M}_{0,r}}$. Their (real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over $\mathcal{M}_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving "first-order" K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.
 Keywords: Schubert calculus, stable curves, Shapiro-Shapiro Conjecture, jeu de taquin, growth diagram, promotion
 MSC Classifications: 14N15 - Classical problems, Schubert calculus 05E99 - None of the above, but in this section

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