There are some discussion of shellable simplicial complexes here Testing simplicial complexes for shellability. My question is the following:
Assume that $\Delta$ is a pure simplicial complex on a finite vertex set $V$. Assume that $\Delta$ is Cohen-Macaulay, i.e., it has at most top-dimensional homology. (There are examples of CM but non-shellable simplicial complexes). Moreover, assume that the symmetric group $S_V$ acts transitively on $\Delta$. Could this possibly imply that the simplicial complex $\Delta$ is shellable?