When are permutation invariants Cohen–Macaulay over all fields?

Abstract

We prove that the polynomial invariants of a permutation group are Cohen–Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The “if” direction of the argument uses Stanley–Reisner theory and a recent result of Christian Lange in orbifold theory. The “only if” direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen–Macaulayness.