In this post, I will discuss a remarkable connection between the ADE diagrams and the regular polyhedra in 3-space (or, more precisely, the finite groups of rotations).
The five regular polyhedra have been central to mathematics since the time of Pythagoras, if not before. But a little numerology will introduce what comes next.
We do not want to distinguish between a polyhedron and its dual; so there are essentially three, the tetrahedron, cube/octahedron, and dodecahedron/icosahedron. They are associated with the triples (2,3,3), (2,3,4) and (2,3,5) of integers, in various ways:
- The 2 refers to the fact that an edge contains 2 vertices and bounds two faces. The other two figures give us the faces and vertex figures of the polyhedra (duality interchanges these).
- If we form the barycentric subdivision of the polyhedron associated with (l,m,n) (dual polyhedra have the same barycentric subdivision), and project it onto the sphere, we obtain triangles with angles π/l, π/m and π/n.
- The rotation group (again, dual polyhedra have the same rotation group) is generated by rotations a,b,c satisfying al = bm = cm = abc = 1; and these are defining relations for the group.
The connection of these integers with the E diagrams is simply that they are the lengths of the three arms of the diagrams E6, E7 and E8 respectively, if we re-define length to be the number of nodes (including the trivalent node) rather than the number of edges.
And just what are these numbers? They are all the solutions of the inequality 1/l+1/m+1/n > 1, where l,m,n are integers greater than 1, and at least two of them are greater than 2 (as required if we are to build a polyhedron).
In fact, the area of a spherical triangle with angles α, β and γ is α+β+γ−π. So, if the barycentric subdivision of a polyhedron has N triangles, then the area of each triangle is 4π/N, and is also π(1/l+1/m+1/n−1); so N = 4/(1/l+1/m+1/n−1). This gives N = 24, 48, 120 in the three cases. It is no coincidence that these are twice the orders of the rotation groups. (Including reflections gives a group twice as large as the rotation group; and it is generated by reflections in the sides of a triangle in the barycentric subdivision.)
There is a 2-to-1 homomorphism onto the group SO3(R) of rotations fixing the origin in real 3-space, from the 2-dimensional complex unitary group SU2(C).
This homomorphism was discovered by Arthur Cayley in 1879, and is clearly explained in the first couple of chapters of Felix Klein’s book on the icosahedron. The rotation group of 3-space acts on the unit sphere, conformally (that is, preserving angles); if we identify the sphere with the Riemann sphere (the complex numbers with a point at infinity), the action is by Möbius (linear fractional) transformations. These linear fractional transformations can be lifted to linear maps on a 2-dimensional vector space with determinant 1; but there is a choice of sign, so two matrices correspond to each rotation.
A more physical way of looking at it is that, if an object in 3-space is rotated once and brought back to its initial position, something has changed. The change can be regarded as the relationship of the object to the rest of the universe; if it were attached to distant points by strings, the strings would become tangled. However, a second rotation in the same direction untangles the strings. This is referred to as Dirac’s “spinor spanner”, and has some connection with the spins of elementary particles, which I don’t understand.
Hold a glass of water in the palm of your hand. Now rotate it once, by bringing your hand under your shoulder and up again. You will be quite uncomfortable! Now rotate again, but moving your hand above your shoulder; miraculously everything is back to normal.
Now it is known that the finite subgroups of SO3(R) fall into two infinite families and three sporadic examples:
- the cyclic group Cn of order n, generated by a rotation through 2π/n;
- the dihedral group D2n, generated by Cn and a rotation reversing the direction of the axis (some authors call this group Dn; it has order 2n);
- the three polyhedral rotation groups, with orders 12, 24 and 60 (they are isomorphic to the alternating group A4, the symmetric group S4, and the alternating group A5).
The dihedral groups, incidentally, correspond to the remaining triples of integers greater than 1 whose reciprocals sum to more than 1, namely (2,2,n). We can regard the dihedral group as the rotation group of a “flat” polyhedron consisting of two regular n-gons stuck back-to-back; we can rotate the figure, or turn it over, swapping the faces. When we take the spherical projection, one face projects onto each hemisphere; each hemisphere is divided into 2n triangles with a common vertex at the pole.
One cannot make a polyhedron-like figure to represent a cyclic group; but the spherical projection consists of 2n lunes, each with a vertex at each pole.
The inverse images of these groups under the homomorphism from SU2(C) to SO3(R) are finite groups with twice the order of the rotation groups, viz. 2n, 4n, 24, 48, 120. These groups are called the binary rotation groups. Since the unitary group contains a unique element of order 2, namely −I, which is in the kernel of the homomorphism, each binary rotation group also has a unique involution.
All the binary rotation groups are described in detail, and explicit matrices given, in Klein’s book.
John McKay is a remarkable mathematician. With Graham Higman, he constructed two of the sporadic simple groups (Janko’s third group and Held’s group), although neither of these constructors is recognised in the popular names for these groups, which celebrate their “discoverers”. (An interesting piece of data for the philosophy of mathematics.)
But his fame rests in part on his ability to notice things that other people missed. Most famous is his equation 1+196883 = 196884, where the numbers on the left are the first two character degrees of the Monster sporadic simple group, and the number on the right is the first non-trivial coefficient of the classical modular function. This observation gave rise to the flourishing subject of monstrous moonshine (maybe the subject of another post).
However, it is a different observation I discuss here. We consider representations of finite groups, linear actions on finite-dimensional vector spaces. One way of obtaining new representations from old is by taking the tensor product and decomposing it. I think it was Burnside who showed that, if G acts faithfully on V, then all irreducible representations of G are obtained by decomposing the tensor powers of V.
Each binary rotation group G comes with a distinguished 2-dimensional representation, arising from its embedding in the 2-dimensional unitary group. Let W be the space affording this representation. Since W is unitary, its tensor square contains the identity representation.
Form a graph as follows: the nodes are the irreducible representations of G; each node is labelled with the dimension of the representation. We join X and Y by an edge if Y is a constituent of the tensor product of X with W. The remark in the preceding paragraph shows that the edges are undirected, and Burnside’s result shows that the graph is connected. Moreover, the dimension of the tensor product of X and W is twice the dimension of X; so the sum of the labels of the neighbours of X is twice the label of X.
Hence, by the characterisation described in the previous post, we have:
The McKay graphs of the binary rotation groups are the extended ADE diagrams.
Unsurprisingly, the correspondence attaches An* to the cyclic group Cn+1; Dn* to the binary dihedral group; and E6*, E7* and E8* to the binary tetrahedral, octahedral and icosahedral groups.
One of the privileges of teaching at Merton College, Oxford was that I taught some extremely able students. One of them was Peter Kronheimer. Towards the end of his first degree, when he was preparing to embark on a doctorate in geometry and mathematical physics, he came to ask me about the McKay correspondence. I explained it as best I could, in roughly the above terms.
His early work was on gravitational instantons, where he extended a hyper-Kähler construction of Hitchin et al. to all the ADE types.
I am afraid I am not competent to explain what he did.
The Poincaré sphere
The Poincaré sphere is a 3-dimensional manifold with the same homology as the usual 3-sphere, but not homeomorphic to it. It can be constructed from 3-space by identifying points based on icosahedral symmetry.
Historically, there is (I understand) some evidence that topologists (including Poincaré) hoped that homology equivalence would imply homeomorphism. The Poincaré sphere refutes this hope; consideration of this example led Poincaré to his famous conjecture, proved recently by Perelman.
Of course we could apply this construction to any finite group of rotations; we obtain a space whose fundamental group is the corresponding binary rotation group. Now homology detects the abelianised fundamental group; and the binary icosahedral group is the only one of the binary rotation groups which is a perfect group, that is, has only the trivial abelian quotient.
There has been some speculation that observations of the cosmic microwave background suggest that the Universe is a Poincaré sphere; these, however, have not been confirmed, and are not generally accepted by cosmologists.