Cyclopermutohedron

• Published in: Proceedings of the Steklov Institute of Mathematics 2015, Vol.288 (1), p.132-144
• Main Author: Panina, Gaiane Yu
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 0081-5438; 1531-8605
• DOI: 10.1134/S0081543815010101

It is well known that the k -faces of the permutohedron Π n can be labeled by (all possible) linearly ordered partitions of the set [ n ] = {1,..., n } into n − k nonempty parts. The incidence relation corresponds to the refinement: a face F contains a face F ′ whenever the label of F ′ refines the label of F . We consider the cell complex CP n +1 defined in a similar way but with the linear ordering replaced by the cyclic ordering. Namely, the k -cells of the complex CP n +1 are labeled by (all possible) cyclically ordered partitions of the set [ n + 1] = {1,..., n + 1} into n + 1 − k > 2 nonempty parts. The incidence relation in CP n +1 again corresponds to the refinement: a cell F contains a cell F ′ whenever the label of F ′ refines the label of F . The complex CP n +1 cannot be represented by a convex polytope, since it is not a combinatorial sphere (not even a combinatorial manifold). However, it can be represented by some virtual polytope (that is, the Minkowski difference of two convex polytopes), which we call a cyclopermutohedron CP n +1 . It is defined explicitly as a weighted Minkowski sum of line segments. Informally, the cyclopermutohedron can be viewed as a “permutohedron with diagonals.” One of the motivations for introducing such an object is that the cyclopermutohedron is a “universal” polytope for moduli spaces of polygonal linkages.