Set-valued tableaux rule for Lascoux polynomials

Abstract

Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.