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 structure and enumeration.

Contribution

It introduces a fast local algorithm for computing the monodromy map on skew tableaux and establishes a bijection with genomic tableaux, linking geometry and $K$-theory combinatorially.

Findings

The algorithm efficiently computes the monodromy operator $oldsymbol{ extomega}$.

A bijection with genomic tableaux links $K$-theoretic coefficients to combinatorial structures.

Purely combinatorial proofs are provided for numerical results in $K$-theory and real geometry.

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 fast, 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. Using this bijection, we give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$.