CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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 Sch├╝tzenberger 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 Schubert calculus, stable curves, Shapiro-Shapiro Conjecture, jeu de taquin, growth diagram, promotion
MSC Classifications: 14N15, 05E99 show english descriptions Classical problems, Schubert calculus
None of the above, but in this section
14N15 - Classical problems, Schubert calculus
05E99 - None of the above, but in this section
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/