Shellable Complexes from Multicomplexes

Suppose a group $G$ acts properly on a simplicial complex $\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\Gamma$ must be a subcomplex of $\Lambda = \Delta^{l-1}* \partial \Delta^{p_1-1}*... * \partial \Delta^{p_m-1}$. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of $\Lambda$. We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes.