Group actions on Stanley-Reisner rings and invariants of permutation groups

Let H be a group of permutations of x₁,..., x_n and let Q^H[x₁, x₂,..., x_n] denote the ring of H-invariant Polynomials in x₁, x₂,..., x_n with rational coefficients. Combinatorial methods for the explicit construction of free bases for Q^H[x₁, x₂,..., x_n] 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.