On permutation polytopes - notions of equivalence

TL;DR

This paper clarifies the concept of effective equivalence among permutation groups, characterizes their geometric properties, and demonstrates that such groups do not always correspond to affinely equivalent polytopes, answering a specific open question.

Contribution

It provides a geometric characterization of effectively equivalent permutation groups and shows they are not necessarily affinely equivalent, addressing an open problem in the field.

Findings

Effective equivalence is clarified and characterized geometrically.

Examples show effectively equivalent groups are not always affinely equivalent.

The paper answers an open question from prior research.

Abstract

We clarify the notion of effective equivalence and characterize geometrically the effectively equivalent permutation groups. In particular, we present examples showing that the latter do not correspond to affinely equivalent polytopes thereby answering Question 2.12 of [BHNP09]. We apply our characterization to our examples and formulate several questions.