Two-Orbit Polytopes

TL;DR

This paper develops a comprehensive structural theory for two-orbit abstract polytopes, extending existing group-theoretic frameworks to classify and analyze their automorphism groups and face configurations.

Contribution

It introduces a general framework for understanding two-orbit polytopes of any rank, including their automorphism groups and flag structures, expanding beyond regular and chiral cases.

Findings

Characterized face- and section-transitivity properties of two-orbit polytopes.

Described automorphism groups via generating sets and stabilizer subgroups.

Provided a classification based on local flag configurations.

Abstract

Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological characteristics, that generalize (the face lattice of) traditional polyhedra, polytopes or tessellations. Most research has focused on abstract polytopes with the highest possible symmetry, in particular those that are regular or chiral. In this paper we study two-orbit polytopes, that is, abstract polytopes whose automorphism groups have exactly two orbits on flags. Such polytopes of rank $n$ fall into $2^n-1$ classes, determined by their local flag configuration. We develop a general structural theory of two-orbit polytopes of arbitrary rank. In particular, we determine their face- and section-transitivity properties and describe the structure of their automorphism groups via distinguished generating sets and face stabilizer subgroups. These results yield a characterization of the partial order { on the polytope} in terms of the automorphism group. Two-orbit polytopes in different classes behave quite differently. Our approach extends the group-theoretic framework for regular and chiral polytopes and provides a systematic foundation for the study of polytopes with two flag orbits.