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.
Categories: math.CO