  We study the equivariant Ehrhart theory of families of polytopes that are
invariant under a non-trivial action of the group with order two. We study
families of polytopes whose equivariant $H^*$-polynomial both succeed and fail
to be effective, in particular, the symmetric edge polytopes of cycles and the
rational cross-polytope. The latter provides a counterexample to the
effectiveness conjecture if the requirement that the vertices of the polytope
have integral coordinates is loosened to allow rational coordinates. Moreover,
we exhibit such a counterexample whose Ehrhart function has period one and
coincides with the Ehrhart function of a lattice polytope.

Combinatorics;Representation Theory