| A group action on a simplicial complex is proper if whenever an element of the group fixes a face of the complex, it fixes that face pointwise. In particular, the orbit of a vertex cannot form a face of the complex unless that vertex is fixed; this places restrictions on which sets of vertices can be contained faces of the complex. Necessary conditions have been found on the number of faces of a Cohen-Macaulay complex subject to these face conditions (but not necessarily having a proper group action), in the first part of this dissertation we show that these conditions are in fact sufficient. In the second part we consider simplicial fans having proper group actions, and show that if the group is cyclic, then the representation of the group carried by the cohomology of the toric variety associated to the fan is a permutation representation. |
