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].

On the Sperner property for the absolute order on complex reflection groups

“…The weak order is defined by an equation similar to (1), but with the reflection length R replaced by the more classical length , which records the shortest decomposition of an element of W as a product of simple reflections. The strong Sperner property for the weak order in type A n was previously established by the authors [6]. Another related order, the (strong) Bruhat order, is known to be strongly Sperner for all Coxeter groups by work of Stanley [17].…”

“…By Equation (6) and the classical result that the coefficients of a real-rooted real polynomial are log-concave, we may apply Theorem 2.5 to see that Abs(W ) admits a normalized flow (with ν ≡ 1), implying that it is strongly Sperner.…”

The Sperner property for 132‐avoiding intervals in the weak order

“…where [e, π] R denotes the interval below π in the right weak order. The operator F is the restriction of an operator suggested by Stanley [16] and E is a significant generalization of the operator used in [4]. Let e, f, h denote the standard generators for the Lie algebra sl 2 (C) (see Section 2).…”

“…Establishing the Sperner property for the weak order on a parabolic quotient would thus be a strengthening of Theorem 1.1. Björner [1] asked whether the weak order on the whole symmetric group had the (strong) Sperner property; it was conjectured by Stanley [16] that it does, and this was proven in [4] by establishing a certain action of the Lie algebra which respects the weak order. This technique was developed further in [3, 5, 6] in order to establish new formulas for principal evaluations of Schubert polynomials.…”

Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley

“…Recently, C. Gaetz and Y. Gao [GG18] proved the invertibility of Stanley's matrix by constructing an action of the Lie algebra sl 2 . We give a new proof of invertibility by proving Stanley's determinant conjecture.…”