A combinatorial 𝔰𝔩₂-action and the Sperner property for the weak order

Abstract

We construct a simple combinatorially-defined representation of s l 2 \mathfrak {sl}_2 which respects the order structure of the weak order on the symmetric group. This is used to prove that the weak order has the strong Sperner property, and is therefore a Peck poset, solving a problem raised by Björner [Orderings of Coxeter groups, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195]; a positive answer to this question had been conjectured by Stanley [Some Schubert shenanigans, preprint, 2017].