A canonical expansion of the product of two Stanley symmetric functions

TL;DR

This paper introduces a natural way to expand the product of two Stanley symmetric functions by analyzing their stabilization as Schubert polynomial products, providing new insights into their structure and stability properties.

Contribution

The paper establishes a stabilization approach for expanding Stanley symmetric function products via Schubert polynomials, offering a new perspective and proof for their expansion.

Findings

Expansion stabilizes as n increases

Better understanding when one permutation is Grassmannian

Provides a second proof of the main result

Abstract

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.