Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths

For a crystallographic root system ${\mathfrak D}$ we consider the simplicial complex $Δ_{\mathfrak D}$ of all antichains in the root poset of ${\mathfrak D}$. We show that $Δ_{\mathfrak D}$ is shellable if and only if ${\mathfrak D}$ is $A_n$, $B_n$, $D_3$ or $G_2$. Since antichains in types $A_n$