One-dimensional Schubert problems with respect to osculating flags

TL;DR

This paper studies one-dimensional Schubert problems with osculating flags on the rational normal curve, revealing their real geometry via tableau operations and establishing a new identity in K-theoretic Littlewood-Richardson coefficients.

Contribution

It extends the understanding of Schubert problems to one-dimensional cases over moduli spaces, connecting real geometry with tableau combinatorics and introducing a novel K-theoretic identity.

Findings

Real points of solution curves are smooth over M_{0,r}

Geometry described by orbits of Schützenberger promotion and evacuation

New identity involving K-theoretic Littlewood-Richardson coefficients

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 $\overline{M_{0,r}}$, allowing the points to collide. These give rise to smooth covers of $\overline{M_{0,r}}(\mathbb{R})$, 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 $\overline{M_{0,r}}$. Their (real) geometry turns out to be described by orbits of Sch\"{u}tzenberger promotion and a related operation involving tableau evacuation. Over $M_{0,r}$, 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.