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

Abstract Let H be a group of permutations of x 1 ,…, x n and let Q H [ x 1 , x 2 ,…, x n ] denote the ring of H -invariant Polynomials in x 1 , x 2 ,…, x n with rational coefficients. Combinatorial methods for the explicit construction of free bases for Q H [ x 1 , x 2 ,…, 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.

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