A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions F _w ?F _u into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_{w}=\lim_{n\to\infty}\mathfrak{S}_{1^{n}\times w}$ , and study the behavior of the expansion of $\mathfrak {S}_{1^{n}\times w}\cdot \mathfrak {S}_{1^{n}\times u}$ into Schubert polynomials as n increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stability properties, providing a second proof of the main result.