Equivariant Hilbert and Ehrhart series under translative group actions

We study representations of finite groups on Stanley–Reisner rings of simplicial complexes and on lattice points in lattice polytopes. The framework of translative group actions allows us to use the theory of proper colorings of simplicial complexes without requiring an explicit coloring to be given. We prove that the equivariant Hilbert series of a Cohen–Macaulay simplicial complex under a translative group action admits a rational expression whose numerator is a positive integer combination of irreducible characters. This implies an analogous rational expression for the equivariant Ehrhart series of a lattice polytope with a unimodular triangulation that is invariant under a translative group action. As an application, we study the equivariant Ehrhart series of alcoved polytopes in the sense of Lam and Postnikov and derive explicit results in the case of order polytopes and of Lipschitz poset polytopes.