# Convex polytopes as “products” of lower dimensional polytopes of the same family

This MO-Q details the sense in which an associahedron is a product of lower dimensional associahedra, and this MSE-Q indicates the same is true for permutohedra.

Is there a reference which classifies the families of convex polytopes for which this relationship holds and provides more detail?

(Edit 6/6/2016 in response to Alen Knutson's comments)

For the n-Dim simplices, the n-D simplex can be generated by connecting the vertices of the (n-1)-D simplex to a point in a dimension orthogonal to those of the (n-1)-D simplex, so each set of k-D faces is associated to one unique simplex and the face polynomial $$\frac{(x+1)^{n+1} - 1}{x}$$ has no refined components.

For the hypercubes, the n-Dim hypercube can be generated by translating the (n-1)-Dim hypercube in an orthogonal direction, so again each set of k-D faces is associated to a unique lower order hypercube and the face polynomial is $$(x+2)^n$$ with no refined components.

The simplices and the hypercubes have a higher degree of symmetry and much simpler geometric constructions than the permutahedra and associahedra, giving an extremely rough classification scheme for the two sets of convex polytopes.

Edit (Dec 2017):

For the permutohedra and associahedra, see pg. 5 of "Hopf monoids and generalized permutahedra" by Aguiar and Ardila in which we find " ... every face of a permutahedron is a product of permutahedra. ... every face of an associahedron is a product of associahedra ... ."

Edit (June 2018):

The permutahedra and associahedra share some common properties. The permutohedra and their simplicial duals share the same symmetric h-vectors as well as the associahedra and their simplicial duals. (I don't know how common or relevant this property is.) Furthermore, Postnikov in Positive Grassmannian and Polyhedral Subdivisions states on pg. 17 the following:

(1) For a projection of the (n-1)-simplex to an n-gon, π-induced subdivisions are exactly the subdivisions of the n-gon by noncrossing chords. All of them are coherent. The fiber polytope (or the secondary polytope) in this case is the Stasheff associahedron.

(2) For a projection of the n-hypercube to a 1-dimensional line segment, the fiber polytope is the permutohedron.

For more notes on fiber polytopes, see Combinatorics of Polytopes by Barvinok.

(Edit Nov. 5, 2018)

From "Higher homotopy operations" by Blanc and Markl:

Definition. A family of polytopes is a sequence $$F = (P_n)^\infty_{n=0}$$ of polytopes, starting with $$P_0 = {pt}$$, such that $$dim(P_n) = n$$, and each facet of $$P_n$$ is isomorphic to some product of lower dimensional polytopes from $$F$$. ... Many familiar examples of polytopes fit into such families:

The authors then list the n-simplices, hypercubes, associahedra, and permutohedra.

Edit (Sept. 2019): In "The diagonal of the associahedra" Masuda, Thomas, Tonks, and Vallette discuss face-coherent fsmilies of convex polytopes such as the simplices, hypercubes, and associahedra.

• Is this saying more than "each face is a product of polytopes from the family"? If it's only that, then I don't see that any sort of classification is possible. – Allen Knutson Jun 4 '16 at 12:23
• @AllenKnutson, no more, no less, as encoded in the subscripts and exponents of the monomials of the refined face partition polynomials. – Tom Copeland Jun 4 '16 at 12:53
• Then take any set of polytopes. Close the set under taking faces. Now it has your property. E.g., take all the permutahedra of dimension $\leq 17$. I think you'll need some positive direction like, if a polytope's facets all are products from the set, plus some extra condition, then that polytope should be in the set. I don't know what this condition would be with the set of associahedra or permutahedra. – Allen Knutson Jun 4 '16 at 22:24
• @AllenKnutson, a related question for me is "Why do refined face partition-polynomials exist that pose a bijection between the faces of these two families of polytopes and the partitions of the integers?" For the permutohedra, I suppose the answer should relate the structure of the polytopes to the Faa di Bruno formula for composing an e.g.f. with 1/x. For the n-simplices, this bijection doesn't hold whereas the geometric constuction of higher dim simplices from lower is simple. – Tom Copeland Jun 4 '16 at 23:23
• For an interesting class of polytopes closed under taking faces, see combinatorics.org/ojs/index.php/eljc/article/view/v11i1r65/pdf. As a special case, faces of cyclic polytopes are cyclic polytopes. – Richard Stanley Jun 26 '18 at 14:47