Happy New Year

And welcome to 2024, the 24th year of the 3rd millenium. By happy coincidence, 2024 is also the number of ways of choosing a triple of 3 objects out of 24, i.e. 2024 = 24 x 23 x 22 / 3 x 2 x 1. The number 24 is itself of very special significance in many parts of mathematics, and also in physics. In particular, it is the order of the binary tetrahedral group that I use to describe all internal symmetries of elementary particles. In some ways of counting, it is also the number of distinct elementary particles, namely 3 x 4 fermions (3 generations of neutrinos, electrons, up and down quarks) plus 1+3+8 bosons (1 photon, 3 intermediate vector bosons and 8 gluons).

The three binary polyhedral groups have orders 24 (tetrahedral), 48 (octahedral) and 120 (icosahedral), which happen to be equal to 5×5-1, 7×7-1 and 11×11-1 respectively. Is this a mere coincidence, or a fact of deep significance? I believe, with McKay, that it is the latter. For example, it enables us to construct some very interesting Frobenius groups, of orders 25×24, 49×48 and 121×120 respectively. The smallest of these cases consists of 25 copies of the binary tetrahedral group (called Frobenius complements), which intersect only in the identity element, together with a group of order 25 (called the Frobenius kernel). You can check that the numbers add up: 25 x (24-1) + 25 = 25 x 24 = 600. Does this Frobenius group have any significance for physics? I doubt it.

A closely related fact about the primes 5, 7 and 11, that distinguishes them from all larger primes, is that you can get the 2×2 matrices written over integers modulo 5, 7, and 11 to act on 5, 7 and 11 objects respectively, where you would normally expect to need 6, 8 and 12 respectively (i.e. p+1 for a prime p). It is also true for the primes 2 and 3 that you can get actions on 2 and 3 objects as well as the more obvious 3 and 4 respectively. The case p=3 here is of immense significance for physics, because this is what creates the three generations of leptons (and presumably also quarks, though I’m not quite sure about that). The reason why physicists do not understand this fact is because they will insist on writing their 2×2 matrices over complex numbers, instead of over finite fields. Hence they miss this exceptional behaviour of the prime 3.

In geometrical terms, the tetrahedron has 3+1=4 vertices which correspond to the generic p+1 points, but also has 3 diagonals joining the midpoints of opposite edges, giving the exceptional set of p objects. The case p=5 can be visualised with an icosahedron, which has p+1=6 pairs of opposite vertices, giving the generic action, but also has a set of 5 inscribed cubes, giving the exceptional action on p objects.

Well now, perhaps you’d like me to tell you something about the significance of the number 2024? As you know, the most interesting group that acts on a set of 24 objects (points, letters, whatever you want to call them) is the Mathieu group M24, discovered by Emile Mathieu (a well known physicist) in 1873. He used the fact that 24=23+1, and the fact that 23 is prime, to start from 2×2 matrices modulo 23 and build M24 out of this. If you have tried to read his papers (as I have) then you will see that the calculations are horrendous, and you will harbour some doubt as to whether they are actually correct. In 1901, G. A. Miller went so far as to publish a “proof” that M24 does not exist, though, to be fair, he quickly retracted this when he realised his mistake. Nevertheless, it was not until 1938 that Witt published a proof of existence of M24 that was simple enough to be completely convincing. Although, if we are to be strictly honest, it is not completely obvious that Witt constructed the same group that Mathieu constructed.

So let me tell you a little bit about Witt’s proof. He split 24 as 21+3, and built up from 21 to 22, from 22 to 23, and from 23 to 24. His proof boils down to showing that all 2024 triples are equivalent: there is one “special” one in his construction, that splits 0+3 across 21+3; then there are 21×3=63 that split 1+2, and 210×3=630 that split 2+1, and after that it gets complicated, because the 3+0 case splits into two subcases. Anyway, he begins by looking at the subgroup that fixes one of the 2024 triples. It is easy to calculate its order, but you need to know quite a lot of group theory to figure out which group it actually is. The question is really, what is the geometric structure of the 21 points? It turns out to be a projective plane of order 4, with (therefore) 4×4 + 4 + 1 = 21 points, and 21 “lines” each with 4+1=5 points on it. So you can study this geometry using 3×3 matrices with entries in the field of order 4. This field has elements 0, 1, v, w with vw=1, so vv=w and ww=v. The determinants of the matrices can be 1, v or w, and these determinants label the transitions between the 3 extra points.

So now he is ready to start, by fixing one of the three extra points, joining it on to the 21 to make 22, and showing that these 22 points are all equivalent (i.e. in mathematical language, the symmetry group is transitive on the 22 points). Then he does the same thing twice more, with the other two points, proving transitivity on 23 points, and then on 24. This gives transitivity on 24x23x22 ordered triples, and therefore on 24x23x22/3x2x1 = 2024 unordered triples. But the really exciting thing is that any two points in the projective plane are equivalent to any other two, so that there is transitivity on the 24x23x22x21x20 ordered quintuples. Once you get to sextuples, however, you lose transitivity, because once you have chosen two points in the projective plane, there are 3 other points on the line joining them, and 16 points not on this line. So the really special property of M24 is that it is quintuply transitive but not sextuply transitive. Assuming you have at least 8 points (to avoid trivialities) there is only one other group that is quintuply transitive but not sextuply transitive, and that is M12, discovered by Mathieu in 1861.

Does M24 have any significance for physics? Somehow I doubt it, but many people have been intrigued by the unique properties of M24, and its lattice cousin the Conway group, and its giant offspring the Monster, and have tried to make connections to physics. The occurrence of 3×3 matrices as a fundamental part of Witt’s construction of M24 is suggestive of some connection to the strong force. The field of order 4 is a finite analogue of the complex numbers, so we get an analogue of 3×3 complex matrices. They are not unitary, but we can restrict to the unitary matrices, which is equivalent to restricting to M12. The unitary group in this case has a normal subgroup of order 9, and a complement which is the binary tetrahedral group. So if you think that the Monster has anything to do with physics, then you really ought to start by looking at the binary tetrahedral group.

There is a “supersymmetry” relating two different actions of M12 on 12 points. In one action the unitary group acts on a decomposition of the 12 points into four disjoint triples, much like the 12 fundamental fermions in the Standard Model. In the other action the unitary group acts as 3+9, or if we restrict to the binary tetrahedral group, as 3+1+8, much like the 12 fundamental bosons in the Standard Model. Is this significant, or is it just a meaningless coincidence of small numbers? Much as I would like to believe it is significant for physics, I don’t. Supersymmetry, as a physical concept uniting fermions and bosons, is well and truly dead, thanks to experiments at the Large Hadron Collider. What it tells us is, in my opinion, not that M12 is important for physics, but that the binary tetrahedral group is important for physics.

I could be wrong, of course.