A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

TL;DR

This paper develops a new jeu de taquin theory for increasing tableaux, enabling combinatorial rules for K-theoretic Schubert calculus on Grassmannians and potentially on all minuscule flag varieties, extending classical combinatorial frameworks.

Contribution

It introduces a novel jeu de taquin framework for increasing tableaux and applies it to derive a new combinatorial rule for K-theoretic Schubert calculus, generalizing previous methods.

Findings

Provides a new combinatorial rule for K-theory Schubert calculus

Extends jeu de taquin to increasing tableaux

Proposes a conjectural uniform rule for minuscule flag varieties

Abstract

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch\"{u}tzenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of [Buch '02] and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending [Thomas-Yong '06]. We also present analogues of results of Fomin, Haiman, Schensted and Sch\"{u}tzenberger.