Set-Valued Skyline Fillings

TL;DR

This paper introduces a set-valued extension of semistandard skyline fillings, unifying them with set-valued tableaux in combinatorial $K$-theory, and establishes a bijection with Grothendieck polynomial expansions.

Contribution

It defines a novel set-valued version of skyline fillings and connects them to set-valued tableaux and Grothendieck polynomials through new bijections.

Findings

Unified set-valued skyline fillings with semistandard tableaux

Established bijection with Schur expansion of Grothendieck polynomials

Extended combinatorial models in $K$-theory

Abstract

Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart's Schur expansion of the Grothendieck polynomial $G_\lambda$, using the uncrowding operator of V. Reiner, B. Tenner, and A. Yong.