||One-dimensional Schubert problems with respect to osculating flags|
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
allowing the points to collide.
These give rise to smooth covers of
$, with structure
and monodromy described by Young tableaux and jeu de taquin.
In this paper, we give analogous results on one-dimensional Schubert
Their (real) geometry turns out to
be described by orbits of SchÃ¼tzenberger promotion and a
related operation involving tableau evacuation. Over
our results show that the real points of the solution curves
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