Key polynomials and a flagged Littlewood-Richardson rule

This paper studies a family of polynomials called key polynomials, introduced by Demazure and investigated combinatorially by Lascoux and Schützenberger. We give two new combinatorial interpretations for these key polynomials and show how they provide the connection between two relatively recent combinatorial expressions for Schubert polynomials. We also give a flagged Littlewood-Richardson rule, an expansion of a flagged skew Schur function as a nonnegative sum of key polynomials.