Let H be a group of permutations of x1,..., xn and let QH[x1, x2,..., xn] denote the ring of H-invariant Polynomials in x1, x2,..., xn with rational coefficients. Combinatorial methods for the explicit construction of free bases for QH[x1, x2,..., xn] as a module over the symmetric polynomials are developed. The methods are developed by studying the action of the symmetric group on the Stanley-Reisner ring of the subset lattice. Some general results are also obtained by studying the action of a Coxeter group on the Stanley-Reisner ring of the corresponding Coxeter complex. In the case of a Weyl group, a purely combinatorial construction of certain invariants first considered by R. Steinberg (Topology 14 (1975), 173-177) is obtained. Some applications to representation theory are also included.