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.

Tropical moduli spaces as symmetric Δ$\Delta$‐complexes

“…Swartz considered the case where the subgroup of the orthogonal group is isomorphic to (ℤ∕2ℤ) 𝑘 [13], and Lange classified the cases where the quotient is a PL-sphere [11] or a topological sphere [12]. See also [3] for applications of Lange's results to quotients of 𝑆 𝑛−1 by subgroups of the permutation group 𝔖 𝑛+1 .…”

A rotation group whose subspace arrangement is not from a real reflection group

“…In the last four decades, bireflection groups have also turned out to have invariant-theoretic significance; see [ES80], [KW82], [Kem99], [LP01], [GK03], [Duf09], [DEK09], [BSM18]. In addition, in the real case, Christian Lange [Lan16] has recently shown that bireflection groups (also known, when working over R, as reflection-rotation groups) are characterized by a remarkable topological property: they are exactly those finite groups G of linear automorphisms of R n such that the orbifold R n {G is a piecewise-linear manifold (possibly with boundary).…”