In the preceding post with this title, I showed how to translate the existence question for regular polytopes into one concerning groups, specifically *string C-groups*. I will begin by saying a bit more about the reverse construction.

Suppose that we have a string C-group *G* of rank *d*, generated by involutions ρ_{i}, for *i* in {0,…,*d*−1}. As earlier, let *G _{S}* be the subgroup generated by the involutions ρ

_{i}for

*i*∈

*S*; and let

*H*be the group

_{i}*G*where

_{S}*S*consists of all the indices except

*i*. Then

*H*is the stabiliser of the flag

_{i}*f*in our standard flag. By transitivity, the other

_{i}*i*-dimensional faces are parametrised by the cosets of

*H*in

_{i}*G*.

So we can take the faces to be the cosets of the subgroups *H _{i}*, for

*i*= 0,…

*d*−1; two faces are incident if the corresponding cosets have non-empty intersection. This recovers the structure of the polytope. By the intersection property,

*G*

_{∅}is the trivial group, and

*G*

_{{i}}is the subgroup of order 2 consisting of the identity and ρ

_{i}.

In our cube example, let us number the vertices from 1 to 8, so that the special vertex *v* is 1, the special edge *e* is 12, and the special face *f* is 1234; let 5,6,7,8 be the vertices on the other face adjacent to 1,2,3,4 respectively. Then

- ρ
_{0}is the reflection swapping 1 and 2, so as permutation it is (1,2)(3,4)(5,6)(7,8). - ρ
_{1}is the reflection in the plane through 15 bisecting the angles between the adjacent faces; so it is (1)(3)(5)(7)(2,4)(6,8); - ρ
_{2}is the reflection in the plane through 12 and the opposite edge 78; so ρ_{2}is (1)(2)(7)(8)(3,6)(4,5).

It is easy to verify that this group has the required properties.

The group can be encoded if we have a faithful permutation representation of it. Suppose that *G* acts faithfully on the set {1,…*n*} for some *n*. Now form the edge-coloured multigraph on this vertex set, in which *x* and *y* are joined by an edge of the *i*th colour if (*x,y*) is a cycle of the permutation corresponding to ρ_{i}. From the edge-coloured multigraph, we can recover the permutations ρ_{i} (their non-trivial cycles are the edges of the *i*th colour) and hence the group *G*.

This graph is called a *CPR graph* (for “C-group permutation representation graph”).

If we take the set on which *G* acts to be the set of maximal chains, we obtain the Cayley graph of *G* with respect to our distinguished generators. Another natural choice for the set is the set of *j*-dimensional faces (if this action is faithful); this is what we did above for the cube, with *j* = 0.

A set *S* of elements of a group *G* is said to be *independent* if no element of *S* is contained in the subgroup generated by the remaining elements. By the Intersection Property, the distinguished generators of the group of a regular polytope are independent.

A theorem of Julius Whiston (discussed here) shows that the largest size of a set of independent elements in the symmetric group *S _{n}* is

*n*−1, and that if equality holds then the independent set generates the symmetric group. Philippe Cara and I found all the independent generating sets of size

*n*−1 in

*S*. The only case in which they are all involutions is where they correspond to the edges of a tree, and the only such case in which we have a string C-group is when the tree is a string (the Coxeter–Dynkin diagram of type

_{n}*A*(as described here). The corresponding polytope is the (

_{n}*n*−1)-simplex (the tetrahedron for

*d*= 3,

*n*= 4).

So we see that a regular polytope having a CPR-graph with *n* vertices must have rank at most *n*−1, with equality if and only if it is a simplex.

Dimitri Leemans and his co-authors have been extending this result, as I hope to describe soon.

In the meantime, let me remark a curious connection with another recent post here. Given a regular polytope of dimension *d*, the subgroups *G _{S}* generated by subsets of the given generators form a lattice isomorphic to the Boolean lattice of rank

*d*. As we saw, the proof of this requires the Intersection Property. If this property does not hold, then we only have a join-semilattice. I discussed this issue here: the existence of a Boolean meet-semilattice of the subgroup lattice of

*G*is equivalent to the existence of a Boolean join-semilattice of the same rank, but not to that of a Boolean lattice of the same rank. I think there are things here deserving further explanation.

To conclude, Marston Conder told me that he and Deborah Oliveros published a paper in the *Journal of Combinatorial Theory* last year, in which they showed that in order to verify that a group generated by involutions does satisfy the Intersection Property, it is not necessary to check all possible intersections.