Cyclopermutohedron

G. Yu. Panina

Informatics and Automation, Geometry, topology, and applications, Tome 288 (2015), pp. 149-162.

Résumé

It is well known that the $k$-faces of the permutohedron $\Pi_n$ can be labeled by (all possible) linearly ordered partitions of the set $[n]=\{1,\dots,n\}$ 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 $\mathrm{CP}_{n+1}$ defined in a similar way but with the linear ordering replaced by the cyclic ordering. Namely, the $k$-cells of the complex $\mathrm{CP}_{n+1}$ are labeled by (all possible) cyclically ordered partitions of the set $[n+1]=\{1,\dots,n+1\}$ into $n+1-k>2$ nonempty parts. The incidence relation in $\mathrm{CP}_{n+1}$ again corresponds to the refinement: a cell $F$ contains a cell $F'$ whenever the label of $F'$ refines the label of $F$. The complex $\mathrm{CP}_{n+1}$ 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 $\mathcal{CP}_{n+1}$. 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.

Bibliographie

[1] Chapoton F., Fomin S., Zelevinsky A., “Polytopal realizations of generalized associahedra”, Can. Math. Bull., 45 (2002), 537–566 | DOI | MR | Zbl

[2] Gelfand I.M., Kapranov M.M., Zelevinsky A.V., Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, 1994 | MR | Zbl

[3] Farber M., Schütz D., “Homology of planar polygon spaces”, Geom. dedicata, 125 (2007), 75–92 | DOI | MR | Zbl

[4] Kapranov M., “The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation”, J. Pure Appl. Algebra, 85:2 (1993), 119–142 | DOI | MR | Zbl

[5] Pukhlikov A.V., Khovanskii A.G., “Konechno-additivnye mery virtualnykh mnogogrannikov”, Algebra i analiz, 4:2 (1992), 161–185 | MR

[6] Panina G.Yu., “Virtualnye mnogogranniki i klassicheskie voprosy geometrii”, Algebra i analiz, 14:5 (2002), 152–170 | MR

[7] Panina G., “New counterexamples to A.D. Alexandrov's hypothesis”, Adv. Geom., 5:2 (2005), 301–317 | DOI | MR | Zbl

[8] Panina G., Moduli space of a planar polygonal linkage: A combinatorial description, E-print, 2012, arXiv: 1209.3241 [math.AT]

[9] Postnikov A., “Permutohedra, associahedra, and beyond”, Int. Math. Res. Not., 2009:6 (2009), 1026–1106 | MR | Zbl

[10] Ziegler G.M., Lectures on polytopes, Grad. Texts Math., 152, Springer, Berlin, 1995 | DOI | MR | Zbl