Lascoux expansion of the product of a Lascoux and a stable Grothendieck

Abstract: This paper gives a tableau formula for expanding the product of a Lascoux polynomial and a stable Grothendieck polynomial into Lascoux polynomials. Lascoux and stable Grothendieck polynomials are inhomogeneous analogues of key polynomials and Stanley symmetric functions, respectively. Our formula refines the K-theoretic Littlewood-Richardson rule of Buch and extends the key expansion of key times Schur established by Haglund, Luoto, Mason, and van Willigenburg. Our proof is combinatorial, relying heavily on a novel row insertion algorithm of Huang, Shimozono and Yu.