On permutation polytopes

TL;DR

This paper explores the combinatorial structure of permutation polytopes, classifies them in low dimensions, and analyzes their faces, providing foundational insights into their geometric and algebraic properties.

Contribution

It offers a complete classification of permutation polytopes and their faces in dimensions 2, 3, and 4, along with an analysis of their face types.

Findings

Complete classification of permutation polytopes in dimensions 2, 3, and 4.

List of combinatorial face types up to dimension four.

Insights into the face structure and group relations of permutation polytopes.

Abstract

A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in dimensions 2,3,4, and the corresponding permutation groups up to a suitable notion of equivalence. We also provide a list of combinatorial types of possibly occuring faces of permutation polytopes up to dimension four.