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 ${\mathrm{\Delta }}_{\mathfrak{D}}$ of all antichains in the root poset of $\mathfrak{D}$. We show that ${\mathrm{\Delta }}_{\mathfrak{D}}$ is shellable if and only if $\mathfrak{D}$ is $A_n$, $B_n$, $D_3$ or $G_2$. Since antichains in types $A_n$ and $B_n$ can be identified with Dyck paths and symmetric Dyck paths, respectively, this yields a simplicial complex on Dyck paths. Indeed, in type $A_n$, shellability can be extended to rational Dyck paths. The $f$- and $h$-triangles then yield statistics on (symmetric/rational) Dyck paths. We determine these statistics for $A_n$ and $B_n$ and leave the case of rational Dyck paths as an open problem.