On Permutation Polytopes - Notions of Equivalence

By assigning a permutation polytope to a group we produce new interesting polytopes. For the effective use of this construction method it is desirable to understand which groups are leading to affine equivalent polytopes. Therefore, the notion of effective equivalence has been introduced [BHNP09]. In this note we clarify the notion of effective equivalence and characterize geometrically the effective equivalence permutation groups. Moreover, we present examples showing that the latter do not correspond to affinely equivalent polytopes. We also apply the characterization to the examples. Last but not least notice that we are also providing a framework for the use of representation theoretic methods in the study of permutation polytopes.