Techniques in equivariant Ehrhart theory (Q6621952)

Let \(G\) be a finite group acting linearly on the lattice \(M'\), let \(P\) be a \(G\)-invariant lattice polytope (i.e., the vertices of \(P\) are in \(M'\)), let \(M\) be a translation of \(M'\) intersected with the affine span of \(P\) to the origin, and consider the induced representation \(\rho : G \to\mathrm{GL}(M)\). Let \(\chi_{ mP }\) denote the permutation character associated to the action of \(G\) on the lattice points in the \(m\)th dilate of \(P\), and let \(R(G)\) be the ring of virtual characters of \(G\). Motivated by the action of a finite group on the cohomology of an invariant hypersurface in a toric variety, \textit{A. Stapledon} [Adv. Math. 226, No. 4, 3622--3654 (2011; Zbl 1218.52014)] introduced virtual characters \(\left\{ \phi_j \right\}_{ j \in \mathbf{N} }\) determined through \N\[\N\sum_{ m \ge 0 } \chi_{ mP } \, t^m = \frac{\sum_j \phi_j \, t^j }{ (1-t) \det[I - \rho t] }\text{ in } R(G)[[t]] \, , \N\]\Nthe \textit{equivariant Ehrhart series} of~\(P\). If one restricts to the action of the trivial group, \(\chi_{ mP }\) is the classical \textit{Ehrhart polynomial} and \(\sum_j \phi_j \, t^j\) is the \textit{Ehrhart \(h^*\)-polynomial} of~\(P\).\N\NThe paper under review provides several techniques for computing equivariant Ehrhart series, including zonotopal tilings, symmetric triangulations, combinatorial interpretations of the \(h^*\)-polynomial, and certificates for the (non-)existence of invariant nondegenerate hypersurfaces. These techniques are applied to several families of polytopes, including hypersimplices, orbit polytopes, and graphic zonotopes.