# A combinatorial duality between the weak and strong Bruhat orders

## TL;DR

This paper explores a duality between the weak and strong Bruhat orders using combinatorial operators, establishing new identities, path counting formulas, and Smith normal form equivalences that deepen understanding of these algebraic structures.

## Contribution

It introduces a raising operator for the strong Bruhat order, dual to Stanley's lowering operator, and proves new identities and formulas linking the two orders.

## Key findings

- Proves a Schubert identity dual to previous work.
- Derives formulas for counting weighted paths in strong order.
- Shows powers of dual operators have identical Smith normal forms.

## Abstract

In recent work, the authors used an order lowering operator $\nabla$, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted $\nabla$ 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 $\Delta$ for the \emph{strong} Bruhat order, which is in many ways dual to $\nabla$. 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 $\nabla$ and $\Delta$ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.