Orbit graphs and face-transitivity of k-orbit polytopes

TL;DR

This paper introduces orbit graphs to classify k-orbit polytopes, determines conditions for i-transitivity, and provides a detailed classification and automorphism group generators for three-orbit polytopes.

Contribution

It presents a novel classification framework for k-orbit polytopes using orbit graphs and explicitly classifies three-orbit polytopes with their automorphism groups.

Findings

Orbit graphs effectively classify k-orbit polytopes.

Conditions for i-transitivity are established.

Explicit automorphism group generators for three-orbit polytopes are provided.

Abstract

The orbit graph of a k-orbit polytope is a graph on k nodes that shows how the flag orbits are related by flag adjacency. Using orbit graphs, we classify k-orbit polytopes and determine when a k-orbit polytope is i-transitive. We then provide an explicit classification of three-orbit polytopes, and we describe a generating set for their automorphism groups.