A combinatorial duality and the Sperner property for the weak order
A poset is Sperner if its largest antichain is no larger than its largest rank. In the 1980's, Stanley used the Hard-Lefschetz Theorem to prove the Sperner property for strong Bruhat orders on Weyl groups. I will describe joint work with Yibo Gao in which we prove Stanley's conjecture that the weak Bruhat order on the symmetric group is also Sperner, by exhibiting a combinatorially-defined representation of sl2 respecting the structure of the weak and strong orders. I will explain how this representation gives rise to a combinatorial duality between the weak and strong Bruhat orders and leads to a strong order analogue of Macdonald's reduced word identity for Schubert polynomials.