A canonical expansion of the product of two Stanley symmetric functions

Abstract

We study the problem of expanding the product of two Stanley symmetric functions F[subscript w]⋅F[subscript u] into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert F[subscript w] = lim[subscript n →∞] S[subscript 1[superscipt n]x w], and study the behavior of the expansion of S[subscript 1[superscript n] x w]⋅S[subscript 1[superscript n] x 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.