A combinatorial duality between the weak and strong Bruhat orders

In recent work, the authors used an order lowering operator $\mathrm{\nabla }$, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted $\mathrm{\nabla }$ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator $\mathrm{\Delta }$ for the \emph{strong} Bruhat order, which is in many ways dual to $\mathrm{\nabla }$. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order. We also show that powers of $\mathrm{\nabla }$ and $\mathrm{\Delta }$ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.