A theorem on polytopes

You know what polygons and polyhedra are. How do we extend their study to higher dimensions?

There are two parts to this question. The first involves incidence geometry: vertices, edges, faces, etc. Here the generalisation is fairly straightforward. A polygon has two types of objects, a polyhedron three; so for an r-dimensional polytope, we take objects of r possible types 0,1,…r−1, with incidence relations between different types satisfying certain conditions. (For convenience we take a single object of type −1 and one of type r incident with everything.)

In a polyhedron, an edge is incident with two vertices and two faces, which are incident with one another; and every vertex or face is contained in a triple of mutually incident objects called a maximal flag. We extend this: every set of mutually incident objects is contained in a maximal flag, having one object of each type. Moreover, a next-to-maximal flag, missing just one type of object, is contained in two maximal flags. Now, if a flag misses two types of object, then the objects of these two types indident with the flag form an incidence structure which is either a polygon (if the two missing types are adjacent) ir complete (if not). For example, in a polyhedron, the edges and faces containing a vertex form a combinatorial polygon; while vertices and faces incident with a common edge are incident with one another).

There are also connectedness conditions which I will not discuss.

I will also not discuss the metric properties (involving distances, angles, etc.) which make a polygon regular, but describe a different take on regularity.

Given a maximal flag F, there are exactly r maximal flags differing from it in just one position. All these are fixed by the stabiliser of F in the group of combinatorial automorphisms. So the order of the group does not exceed the number of maximal flags. We say that the polygon is regular if equality holds, which means that the group acts transitively (and regularly) on the set of maximal flags.

In a regular polygon, there is a unique element s_i mapping F to the flag F_i differing from it in position i only; it maps F_i back to F, so its square is the identity. Connectedness implies that these r elements generate the automorphism group. Moreover, if |i−j|>1, then s_i and s_j commute. So the automorphism group of a regular polytope is what is called a string C-group, which includes an intersection condition implying that the intersection of the subgroups generated by the involutions s_i for i∈I and for i∈J is generated by those with i∈I∩J.

Thus the search for regular polytopes “reduces” to the search for representations of a group as a string C-group; and a string C-group generating a group G determines a unique regulqr polytope up to isomorphism and duality.

The simplest family of regular polytopes to describe consists of the simplices (extending the sequence triangle, tetrahedron …). In this case the r generators are the adjacent transpositions in the symmetric group S_n, with n = r+1, occurring in the representation of this group as a Coxeter group of type A_n (discovered by E. H. Moore in 1896).

Maria Elisa Fernandes and Dimitri Leemans investigated regular polytopes with automorphism group S_n. They showed that there is only one of rank n−1 for n≥5 (the simplex), and, up to duality, only one of rank n−2 for n≥7. Moreover, there are examples of every possible rank from 1 to n−1.

Over the last several years, I have collaborated in work on regular polytopes, first with Dimitri, and then with Maria Elisa, on polytopes whose group is a (transitive) proper subgroup of S_n. We showed that these can have rank at most n/2+c. (For our strugle with the alternating groups, see here.) Now, exploiting these results and the techniques used in their proofs, we have shown what I think a remarkable theorem, proving a conjecture based on the first two results in the preceding paragraph and extensions of them.

Theorem. Suppose that n≥2κ+3. Then the number (up to isomorphism and duality) of regular polytopes of rank n−κ with group S_n depends only on κ, not on n.

(Thus these numbers can be computed by just computing these polytopes of rank r with group S_n with n=2r−3.) The numbers for κ = 1,2,… are 1, 1, 7, 9, 35, 48; we are computing the next number but it is a slow process.

I won’t begin to give the proof here; the paper is over 40 pages long and depends on results from our earlier papers and much else besides. I am very grateful to my coauthors for including me in the project. You can read the paper at . I will just say that, if the inequality between degree and rank holds, we show that there is a weak point in the diagram (called a “perfect split”) where one can insert a transposition to increase both degree and rank by 1.