1. CJM 2016 (vol 69 pp. 143)
||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
2. CJM 2010 (vol 62 pp. 1246)
||Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences|
We prove that the quantum cohomology ring of any minuscule or
cominuscule homogeneous space, specialized at $q=1$, is semisimple.
This implies that complex conjugation defines an algebra automorphism
of the quantum cohomology ring localized at the quantum
parameter. We check that this involution coincides with the strange
duality defined in our previous article. We deduce Vafa--Intriligator type
formulas for the Gromov--Witten invariants.
Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class