A Bijection Between K-Kohnert Diagrams and Reverse Set-Valued Tableaux

Abstract

Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (RSVT) rule for Lascoux polynomials and reverse semistandard Young tableaux (RSSYT) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with 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 RSVT and K-Kohnert diagrams.

Top-degree components of Grothendieck and Lascoux polynomials

“…This rule was first conjectured by Ross and Yong [21]. Notice that our convention is different from [17]: row 1 is the top most row in this paper while it is the bottom most row in [17].…”

“…Let KKDpαq be the set of all K-Kohnert diagrams of α. As proved in [17], K-Kohnert diagrams give a formula for Lascoux polynomials. This rule was first conjectured by Ross and Yong [21].…”