Affine Symmetries of Orbit Polytopes

TL;DR

This paper develops a comprehensive theory of affine symmetries of orbit polytopes, characterizes generic cases, and applies the results to representation polytopes, providing methods to compute symmetries and counterexamples to existing conjectures.

Contribution

It introduces a general framework for understanding affine symmetry groups of orbit polytopes, including generic cases and growth scenarios, and applies this to representation polytopes to compute symmetries effectively.

Findings

Generic orbit polytopes have conjugate affine symmetry groups.

The affine symmetry group equals the original group under certain conditions.

Counterexamples to a conjecture on permutation polytopes are constructed.

Abstract

An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again $G$ if $G$ is itself the affine symmetry group of some orbit polytope, or if $G$ is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows. We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math. 222 (2009), 431--452, Conjecture~5.4].