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 $\mathfrak{S}_{1^n\times w}\cdot\mathfrak {S}_{1^n\times u}$ 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.

05E05, 05C05, 05C35

Schubert polynomials, Stanley symmetric functions, maximal transition tree