Group actions on Stanley-Reisner and Stanley rings

A group G acting on a simplicial complex or, more generally a simplicial poset gives rise to an action on its Stanley-Reisner ring or its Stanley ring, respectively. These rings being both graded, have G-representations on each graded piece of their respective rings. We are concerned with describing all of these G-representations compactly. We will use a result of De Concini, Eisenbud, and Procesi to produce a canonical homogeneous system of parameters, for these rings that is always G-invariant. In this talk we will introduce a new combinatorial invariant, give results for any Cohen-Macaulay simplicial poset, discuss the relationship between this combinatorial invariant and the singular reduced homology of a simplicial complex, and give examples.