When I was a child, one of the radio serials we listened to was The Lone Ranger. Typically, when the Lone Ranger with his sidekick Tonto had cleared up another mess of wrongdoing or whatever, he would say to Tonto, “Our work here is done”, and they would ride off into the sunset. At that moment, not everything would be smoothed out; he would leave the last bit to others. This was because, obviously, he didn’t want to appear in court, when the judge might order him to remove his mask (a big no-no for superheroes!); but also, like Bob Dylan’s John Wesley Harding a decade or so later, his methods were not necessarily entirely above board.
Last Friday, I awoke thinking “Our work here is done”. It took me a while to remember where that line came from; but it turns out to be quite appropriate.
In three earlier posts about regular polytopes (the first one here), I explained how finding an abstract regular polytope with given automorphism group G can be translated into a purely group-theoretic property of G, that it should be a so-called string C-group: this means that G can be generated by a set S of involutions with the two properties
- the involutions can be arranged in a sequence so that two involutions which are not adjacent in the sequence commute (the string property); and
- if U and V are subsets of S, then the intersection of the subgroups generated by U and V is the subgroup generated by the intersection U∩V (the intersection property).
So an interesting question is: Given a finite group G, describe all the regular polytopes with automorphism group G; or, at least, describe the largest possible rank of such a polytope. (The rank of a polytope corresponds, in nice geometric cases, to the dimension of the space in which it is embedded; in the string C-group formulation, it is the size of the generating set S.)
This question was examined, among other people, by Maria Elisa Fernandes, Dimitri Leemans, and Mark Mixer. It is not hard to show that the largest possible rank for the symmetric group Sn is n−1; apart from a few small cases, equality is realised only by the regular simplex of dimension n−1, and the corresponding generating set consists of the Coxeter generators
(1,2), (2,3), …, (n−1,n).
These three authors also did more. They made a remarkable conjecture about polytopes for the symmetric group:
There exists a function f with the property that, for sufficiently large n, the number of different regular polytopes of rank n−k for the symmetric group Sn is f(k).
They proved this conjecture for k = 1,2,3,4, with the values of f(1),…,f(4) being 1, 1, 7, 9. On the basis of computational results, they conjectured that the next value f(5) is equal to 35, and that “sufficiently large” means n ≥ 2k+3.
They also went on to consider regular polytopes whose group is the alternating group An. The largest rank they could find was much smaller, namely ⌊(n−1)/2⌋ for n > 11. (The group A11, exceptionally, is the automorphism group of a regular polytope of rank 6.)
After a very hard month of work in Aveiro, we believe we have a proof of the theorem that this is indeed the largest possible rank for a regular polytope for the alternating group.
I won’t describe the proof here. Suffice to say that we went down to the wire. On Friday, just before lunch we discovered a problem with the penultimate lemma in the paper; not a serious problem, just a lapse of attention which can easily be fixed, but we didn’t have time to fix it that day. The very last lemma, like a number of other results in the paper, also has a small problem in that its proof requires n to be “sufficiently large” (this typically means n > 16 or so; smaller values can be handled either by computer or by being just a little more careful in the proofs.
Assuming it stands up, it is a big theorem. Apart from the symmetric group, most results of this sort have concerned groups of Lie type of very low rank, where the subgroup lattice is much less complicated than that of the alternating group. (Note that, if G is the automorphism group of a regular polytope of rank r, then the subgroup lattice of G contains the Boolean lattice of subsets of an r-set as a sublattice, in a very special way.)
Unlike the Lone Ranger, riding off without a backward glance, I cannot escape the next stages: producing a manuscript, checking it carefully, and so on. But I think the main job is done. Dimitri flew off to Durban yesterday, and I make a smaller journey, by train to Lisbon, today.