## Regular polytopes, 1

One of the topics I am thinking about with Dimitri Leemans at present concerns regular polytopes. He and his co-authors Maria Elisa Fernandes and Mark Mixer have produced some nice results and a tantalising problem about these objects. I will give a brief introduction here; hopefully later I will have some more to report on.

A polytope is a higher-dimensional generalisation of a polygon in 2 dimensions or a polyhedron in 3 dimensions. Rather than stretch your geometric intuition, I will describe polytopes combinatorially. Keep the cube in mind as an example. Here is the cube; I have selected a particular vertex v, edge e and face f. (Ignore the primed vertices for the moment.)

The faces (of whatever dimension) are partially ordered by geometric incidence. (In the cube, a vertex lies on an edge, an edge on a face, or a vertex on a face.) We “complete” the cube by a minimal and maximal element, corresponding to the empty set and the entire polytope.

The polytope, as partially ordered set, has the following properties:

1. Every maximal chain has the same length. (In the case of the cube, a maximal chain has the form (empty set, vertex, edge, face, cube).) So we can talk about the dimension of a face: the empty set has dimension −1, a vertex dimension 0, an edge dimension 1, and so on.
2. A connectedness condition: we can move from any face to any other by a sequence of steps in which consecutive faces are incident; we can further assume that every face in the sequence except the first and last has dimension i or j, where i and j are two given dimensions.
3. If f and g are incident faces of dimensions i and i+2 respectively, then there are exactly two faces of dimension i+1 incident with both f and g. (In our picture of the cube, v and v’ are the faces incident with the empty set and e; e and e’ are incident with v and f; and f and f’ are incident with e and with the whole polytope.)

An automorphism of a polytope is a permutation of the faces preserving the partial order (and hence preserving the dimensions of faces). The collection of all automorphisms is closed under composition and so forms a group, the automorphism group of the polytope. The automorphism group of the cube is, as you would expect, the group S4×C2 of order 48.

It follows from the three conditions above that the identity is the only automorphism fixing a maximal chain. (In the case of the cube, suppose that an automorphism fixes v, e and f. Then it must fix v’, the only other vertex incident with e; similarly it must fix e’ and f’. Using the connectedness, we can work from any maximal chain to any other, and find that everything is fixed.)

So the number of automorphisms does not exceed (and, indeed, is a divisor of) the number of maximal chains. The most symmetric polytopes are thus the ones in which the number of automorphisms is equal to the number of maximal chains, and so the group of automorphisms acts transitively on the maximal chains. These are the regular polytopes.

Suppose that P is a regular polytope. Then there is a unique automorphism of P mapping any maximal chain to any other. We fix a maximal chain C = (f−1,…fd). Now, for any i with 0 ≤ d−1, there is a unique maximal chain Ci which agrees with C in every dimension except i, and so a unique automorphism ρi which maps C to Ci. Then ρi also maps Ci back to C, and so ρi2 = 1, where 1 denotes the identity automorphism.

For example, in the cube, ρ0 reflects the cube in the plane of symmetry bisecting the edge e; ρ1 reflects the cube in the plane of symmetry through v bisecting the face f; and ρ2 reflects the cube in the plane of symmetry through e and bisecting the angle between the faces f and f’.

A connectedness argument shows that, using these reflections in a suitable sequence, we can map C to any maximal chain. So the group G generated by the automorphisms ρi acts transitively on the maximal chains, and so must be equal to the automorphism group of the polytope.

So the automorphism group is generated by d involutions (elements of order 2).

These involutions have two more important properties:

• If |i−j| ≥ 2, then ρi and ρj commute, so their product has order 2. (For example, in the cube, ρ0 and ρ2 are reflections in perpendicular planes.)
• For any subset S of {0,…d−1}, let GS denote the subgroup of G generated by the elements ρi with iS. Then the intersection of GS and GT is equal to GST. This is called the intersection property.

A group generated by involutions with this property is called a string C-group. (“String” because we can imagine the involutions ρ0,…ρd−1 arranged along a string, so that non-adjacent involutions commute; the convention for Coxeter graphs is that involutions are joined by an edge if and only if they do not commute. The “C” stands for “Coxeter”.)

Conversely, any string C-group can be shown to be the automorphism group of a regular polytope.

This material is discussed in a paper by Daniel Pellicer, “CPR graphs and regular polytopes”, in the European Journal of Combinatorics 29 (2008), 59–71.

In the next part, I hope to discuss how we represent and recognise string C-groups, and how this contributes to the theory of regular polytopes.