Monodromy and K-theory of Schubert curves via generalized jeu de taquin

TL;DR

This paper connects the real geometry and K-theory of complex Schubert curves through combinatorial algorithms involving jeu de taquin and promotion, revealing new insights into their monodromy and enumerative properties.

Contribution

It introduces a local algorithm to compute the monodromy operator on skew tableaux and links it to K-theoretic Littlewood-Richardson coefficients, advancing understanding of Schubert curves.

Findings

A new local algorithm for computing the monodromy operator  without rectification.

Bijection between steps in the algorithm and genomic tableaux enumerating K-theoretic coefficients.

Purely combinatorial proofs of numerical results relating K-theory and real geometry of Schubert curves.

Abstract

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(\lambda_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $\omega$ as the monodromy operator. We provide a local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$.