A bijection between $K$-Kohnert diagrams and reverse set-valued tableaux

TL;DR

This paper proves a conjecture by establishing a bijection between reverse set-valued tableaux and $K$-Kohnert diagrams, linking combinatorial models of Lascoux polynomials.

Contribution

It constructs a weight-preserving bijection between $ ext{RSVT}$ and $K$-Kohnert diagrams, confirming a conjecture about combinatorial models for Lascoux polynomials.

Findings

Established a bijection between $ ext{RSVT}$ and $K$-Kohnert diagrams.

Confirmed the conjecture by Ross and Yong on $K$-Kohnert diagrams for Lascoux polynomials.

Provided a combinatorial rule for Lascoux polynomials via $K$-Kohnert diagrams.

Abstract

Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young tableaux ($\mathsf{RSSYT}$) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with $\mathsf{RSSYT}$. Ross and Yong introduced $K$-Kohnert diagrams, which are analogues of Kohnert diagrams. They conjectured a $K$-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between $\mathsf{RSVT}$ and $K$-Kohnert diagrams.