When are permutation invariants Cohen-Macaulay over all fields?

TL;DR

This paper characterizes exactly when polynomial invariants of permutation groups are Cohen-Macaulay over all fields, linking group generators to algebraic properties using advanced combinatorial and geometric methods.

Contribution

It provides a complete characterization of permutation groups with Cohen-Macaulay polynomial invariants over all fields, unifying previous results and introducing new algebraic and geometric techniques.

Findings

Polynomial invariants are Cohen-Macaulay iff the group is generated by transpositions, double transpositions, and 3-cycles.

Uses Stanley-Reisner theory and orbifold theory to prove the 'if' part.

Analyzes inertia groups and employs a local-global approach for the 'only-if' part.

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.