Cosmin Pohoata and Daniel G. Zhu: Hypergraphic Zonotopes and Acyclohedra

I would like to draw your attention to the short beautiful paper Hypergraphic Zonotopes and Acyclohedra by Cosmin Pohoata and Daniel G. Zhu. The paper introduces higher-uniformity analogue of graphic zonotopes and permutohedra, and provides formulas for their volume, and, more generally, for their  Ehrhart polynomials. These formulas are related to the notions of hypertrees and their enumeration that I studied in the early 80s and were discussed in some of the first posts in the blog (I, II, III, IV,V). Cosmin and Daniel also connect the hypergraphic zonotopes to high-dimensional analogs of acyclic orientations of graphs introduced by Linial and Morgenstern.

The graphic zonotopes and permutahedron

Let be the standard basis for . Given a graph with vertices , the graphic zonotope of is the sum of all line segments , where  is adjacent to , and . If is connected, the graphic zonotope ia a -dimensional polytope, and remarkably its volume is the number of spanning trees of . The graphic zonotope of the complete graph is the permutahedron – the convex hull of all permutations of the vector .

Hypergraphs and Hypertrees

Consider now a 3-uniform hypergraph with vertices . We can associate  to the sum of line segments of the form  , where  the form a basis to an dimensional space, and the sum is over all edges of ().  This zonotope is called the hypergraphic zonotope of and when is the complete 3-uniform hypergraph it is called an acyclohedron.  Remarkably the volume of the acyclohedron is a weighted sum of all 2-hypertrees (acyclic spanning subcomplexes with a complete 1-skeleton). The weight for a hypertree is the size of the torsion group . (These formulas extend to -uniform hypergraphs for every .)

Hypertournaments

The vertices of the acyclohedron are in 1-1 correspondence to  the high-dimensional analogs of tournaments introduced in 2013 by Nati Linial and Avraham Morgenstern. We consider orientations of the faces of the complete uniform hypergraph with vertices. A “tournament” refers to an orientation that does not support any element of . Here  refers to the set of nonnegative real numbers.

Remark

The volume of the acyclohedron corresponding to (d+1)-uniform hypergraphs on vertices is

,

where the sum is over all -dimensional simplicial complexes on labelled vertices, with a complete -dimensional skeleton, and which are -acyclic, namely all their (reduced) homology groups with rational coefficients vanish. It will be interesting to give good estimates for this expression. .

As noted by Cosmin and Daniel, these weights are closely related but not identical to those appearing in an old result of mine.  I proved in the early 80s an extension of Cayley’s theorem that refers to the sum of squares of the sizes of the torsion groups (again over the same subcomplexes):

Daniel G. Zhu, Cosmin Pohoata, me, and Arthur Cayley.