A combinatorial ${\mathfrak{sl}}_2$-action and the Sperner property for the weak order

Abstract

We construct a simple combinatorially-defined representation of ${\mathfrak{sl}}_2$ which respects the order structure of the weak order on the symmetric group. This is used to resolve a conjecture of Stanley that the weak order has the strong Sperner property, and is therefore a Peck poset.

6 pages; v2: added conjecture and updated references

Cited by in corpus (4)

• Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
• A combinatorial duality between the weak and strong Bruhat orders
• Padded Schubert polynomials and weighted enumeration of Bruhat chains
• The Sperner property for $132$-avoiding intervals in the weak order