Products of abstract polytopes

TL;DR

This paper extends known polytope products to abstract polytopes, introduces a new topological product, and analyzes their automorphism and monodromy groups, revealing their structural properties and factorizations.

Contribution

It generalizes classical polytope products to the abstract setting, introduces a novel topological product, and studies their automorphism and monodromy groups.

Findings

Unique prime factorization theorems for these products

Automorphism groups expressed in terms of factors' automorphism groups

Products are rarely regular or two-orbit polytopes

Abstract

Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group of a product, show that such a group is always an extension of a symmetric group, and give some examples in which this extension splits.