On orbits of antichains of positive roots

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