The direct sum map on Grassmannians and jeu de taquin for increasing tableaux

TL;DR

This paper explores the geometric and combinatorial aspects of Grassmannian maps, connecting splitting coefficients with Schubert structure constants, and extends jeu de taquin theory to new contexts involving increasing tableaux.

Contribution

It introduces a geometric explanation for splitting coefficients, extends jeu de taquin to increasing tableaux, and provides combinatorial rules for computing various coefficients.

Findings

Established a geometric interpretation of splitting coefficients.

Extended jeu de taquin to increasing tableaux.

Provided rules for computing multiple types of coefficients.

Abstract

The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from [Buch '02] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [Thomas-Yong '09] by proving transparent analogues of [Sch\"{u}tzenberger '77]'s fundamental theorems on well-definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients.