Symmetry Type Graphs of Abstract Polytopes and Maniplexes

TL;DR

This paper extends symmetry type graphs to maniplexes and polytopes, classifies certain orbit types, and provides automorphism group generators, advancing understanding of symmetry in high-dimensional combinatorial structures.

Contribution

It introduces symmetry type graphs for maniplexes and polytopes, classifies 3- and 4-orbit maniplexes, and determines automorphism group generators based on these graphs.

Findings

No fully-transitive $k$-orbit 3-maniplexes with odd $k > 1$ exist.

Classified all 3-orbit maniplexes and face transitivity for 3- and 4-orbit cases.

Provided generators for automorphism groups from symmetry type graphs.

Abstract

A $k$-orbit maniplex is one that has $k$ orbits of flags under the action of its automorphism group. In this paper we extend the notion of symmetry type graphs of maps to that of maniplexes and polytopes and make use of them to study $k$-orbit maniplexes, as well as fully-transitive 3-maniplexes. In particular, we show that there are no fully-transtive $k$-orbit 3-mainplexes with $k > 1$ an odd number, we classify 3-orbit mainplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a polytope or a maniplex, given its symmetry type graph. Finally, we extend these notions to oriented polytopes, in particular we classify oriented 2-orbit maniplexes and give generators for their orientation preserving automorphism group.