Permutation Polytopes of Cyclic Groups

TL;DR

This paper explores the combinatorial and geometric properties of permutation polytopes derived from cyclic groups, providing formulas for their dimensions and vertex degrees, and describing their structure in specific orbit cases.

Contribution

It offers new formulas for dimensions and vertex degrees of permutation polytopes and characterizes their structure for groups with up to two orbits, revealing exponential facet complexity.

Findings

Formulas for dimension and vertex degree of permutation polytopes

Complete combinatorial description for groups with up to two orbits

Existence of exponentially many facets in certain cases

Abstract

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.