A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions $F_w\cdot F_u$ 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 $\s_{1^n\times w}\cdot\s_{1^n\times u}$ 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 stable properties, which provides a second proof of the main result.