Earlier this semester, we had a beautiful seminar by Pierre-Philippe Dechant, whose work has thrown some entirely new light on this beautiful work of art. I would like to explain a bit about this here. For more details, see his papers, for example,
- Pierre-Philippe Dechant, A 3D spinorial view of 4D exceptional phenomena, in Symmetries in Graphs and Polytopes (ed. J. Širáň and R. Jaycay), Springer, 2016, pp. 81-95.
- Pierre-Philippe Dechant, The E8 geometry from a Clifford perspective, Advances in Applied Clifford Algebras (2016).
- Pierre-Philippe Dechant, The birth of E8 out of the spinors of the icosahedron, Proc. Roy. Soc. (A) 472 (2016).
Pierre has done much more than simply add to the literature on the ADE affair. As he puts it in one of his papers,
[The] Lie-centric view point of much of theoretical physics … has unduly neglected the non-crystallographic groups.
Some of his work is well outside pure mathematics, on viruses which have icosahedral symmetry, of which there are many. Why? I never really thought about this before. But a virus has only a tiny amount of genetic material compared to us, and has to reproduce itself using components which are very simple to specify and to assemble: making an icosahedron out of triangles is a good solution to this problem!
The story so far
The ADE affair refers, loosely, to mathematical classification problems where the answer is “two infinite families and three isolated examples”. These come in two main flavours:
- Root systems (with all roots of the same length) and related things such as root lattices, Lie algebras, and cluster algebras: the root systems are of type An, Dn, E6, E7 and E8.
- The 3-dimensional finite rotation groups, and associated reflection groups: the rotation groups are the cyclic and dihedral groups together with the three groups of the regular polyhedra (tetrahedron, octahedron/cube, and icosahedron/dodecahedron).
The relation between the two flavours had been known for some time, but a precise form of it was given by John McKay, who pointed out that the extended Coxeter–Dynkin diagrams associated with the simple Lie algebras coincide with the representation graphs of the binary groups (the double covers of the 3-dimensional rotation groups, pullbacks under the two-to-one isomorphism from the 2-dimensional unitary group to the 3-dimensional orthogonal group).
Precise it may be, but it is rather inexplicit and lacking in explanatory power. Can we do better?
We are dealing here with real Clifford algebras; the complications in characteristic 2 will not affect things.
Pierre Dechant describes a Clifford algebra as a kind of deformation of the exterior algebra by a quadratic form.
The exterior algebra is the largest anticommutative algebra generated by a given vector space V. That is, it is spanned by all products e1∧…∧er, where e1, … er are chosen from a basis for V. The anticommutative law w∧v = −v∧w, and its consequence v∧v = 0, imply that we can start with an ordering of the basis and take e1, …, er in increasing order. So the dimension of the exterior algebra is equal to the number of subsets of the basis, that is, 2n if dim(V) = n.
Now suppose that there is an inner product · on V. The Clifford algebra has the same underlying vector space as the exterior algebra, but the multiplication is given by
vw = v·w+v∧w
on V, extended to the whole algebra. The symmetric and antisymmetric parts of the product are given by the inner product and exterior product, and can be recovered from the Clifford algebra product.
Reflections on V have a particularly simple form in the Clifford algebra; the reflection in the hyperplane perpendicular to a is the map taking v to −ava. Note that the reflections defined by a and −a are the same, so that the group they generate in the Clifford algebra is a double cover of the reflection group they generate on V. The entire group generated by these maps on the Clifford algebra is the Pin group of v, and the corresponding cover of the special orthogonal group (consisting of products of even numbers of reflections) is the Spin group.
So the 120 elements of the group H3 of rotations and reflections of the icosahedron give us 240 elements of the 8-dimensional Clifford algebra; these are, amazingly, the roots in the root system E8.
And more …
There is much more to say, though I won’t say it right now; I might try to persuade Philippe to write it up himself at some point.
For example, the 8-dimensional Clifford algebra is a direct sum of even and odd parts, each 4-dimensional; there is a folding of the E8 root system onto H4 in 4 dimensions. There is also a process called spinor induction which lifts from spinors on 3-space to vectors in 4-space. The correspondence works for other root systems and reflection groups too, forming a 3×3 array, which (row by row) looks like
(A3, B3, H3) → (D4, F4, H4) → (E6, E7, E8).
For now I simply say: Take a look at Philippe’s papers if you want more of this story!