Regular polytopes, 2

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 ρi, for i in {0,…,d−1}. As earlier, let GS be the subgroup generated by the involutions ρi for iS; and let Hi be the group GS where S consists of all the indices except i. Then Hi is the stabiliser of the flag fi in our standard flag. By transitivity, the other i-dimensional faces are parametrised by the cosets of Hi in G.

So we can take the faces to be the cosets of the subgroups Hi, 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 ρi.

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

  • ρ0 is the reflection swapping 1 and 2, so as permutation it is (1,2)(3,4)(5,6)(7,8).
  • ρ1 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);
  • ρ2 is the reflection in the plane through 12 and the opposite edge 78; so ρ2 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 ith colour if (x,y) is a cycle of the permutation corresponding to ρi. From the edge-coloured multigraph, we can recover the permutations ρi (their non-trivial cycles are the edges of the ith 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 Sn 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 Sn. 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 An (as described here). The corresponding polytope is the (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 GS 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.

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About Peter Cameron

I count all the things that need to be counted.
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