On orbits of antichains of positive roots

For any finite poset P , there is a natural operator, X = X P , acting on the set of antichains of P . We discuss conjectural properties of X for some graded posets associated with irreducible root systems. In particular, if Δ + is the set of positive roots and Π is the set of simple roots in Δ + , then we consider the cases P = Δ + and P = Δ + ∖ Π . For the root system of type A n , we consider an X -invariant integer-valued function on the set of antichains of Δ + and establish some properties of it.