There are some discussion of shellable simplicial complexes here 
[Testing simplicial complexes for shellability](https://mathoverflow.net/questions/121371/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?



Cohen-Macaulay versus shellable simplicial complexes
The Dissertation of Frank Lutz
(<http://page.math.tu-berlin.de/~lutz/dissertation.ps>‎) has examples of
Cohen-Macaulay vertex-transitive simplicial complexes that are not
shellable. In particular, there is such a triangulation of Poincaré
dodecahedral space $X$ with 17 vertices. This triangulation is
Cohen-Macaulay since $X$ is a topological 3-manifold with the same
homology groups as a 3-sphere. It is not shellable because it is not
simply-connected.


