A combinatorial duality between the weak and strong Bruhat orders

In recent work, the authors used an order lowering operator del, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted del 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 Delta for the strong Bruhat order on the symmetric group, which is in many ways dual to del. 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, providing a strong order analog of Macdonald's reduced word identity. We also show that powers of del and Delta have the same Smith normal forms, which we describe explicitly, answering a question of Stanley. (C) 2019 Elsevier Inc. All rights reserved.