On orbits of antichains of positive roots

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