Authors: Barbara Baumeister ¹; Christian Haase ²; Benjamin Nill ³; Andreas Paffenholz ⁴

• 1 Universität Bielefeld
• 2 Goethe-Universität Frankfurt am Main
• 3 Case Western Reserve University [Cleveland]
• 4 Darmstadt University of Technology [Darmstadt]

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. 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.