
Onedimensional 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
zerodimensional 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 onedimensional Schubert
problems over
$
\overline{\mathcal{M}_{0,r}}
$.
Their (real) geometry turns out to
be described by orbits of SchÃ¼tzenberger 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 "firstorder" Ktheoretic
LittlewoodRichardson coefficients, for which there does not
appear to be a known combinatorial proof.
Keywords:Schubert calculus, stable curves, ShapiroShapiro Conjecture, jeu de taquin, growth diagram, promotion Categories:14N15, 05E99 