From the 120-cell to E8 (Part 3)

In the last post, we saw how the cell-centres of the 120-cell in 4 dimensions generate a lattice in 8 dimensions. So far, just about the only thing we know about the lattice is that each lattice-point is surrounded by 240 nearest neighbours, each at distance 1.

The 240 nearest neighbours of the origin constitute what is known as a root system. (See the 'Root System' wiki to learn what is involved in this, in general.) We will not attmept to prove it. We only note that the dot-product of any two members of the system is either ½, 0, -½ or -1 (+1 only occurs when a root is dotted with itself). Given a root, one root (itself) has dot product 1 with it, 56 have dot-product ½, 126 have dot-product 0, 56 have dot-product -½, and one (its negative) has dot -1. If you enjoy a bit of mathematical pottering (like me) you may like to pick a root and verify that the system is closed under reflection in the hyperplane through the origin, orthogonal to your chosen root. Thus the roots with dot-product ½ are in 1-to-1 correspondence with the roots whose dot-product is -½.

Given a root system, we can find a set of simple roots, out of which the other roots may be formed as linear combinations with integer coefficients, as follows. The first step is to list all the roots in 'alphabetical order'. In this case, the alphabet consists of the five 'letters' -1, -½, 0, ½, 1, and we regard each root as an 8-letter 'word' (each coordinate being interpreted as a letter). So in the list, all the words beginning with the letter -1 (actually, there is only one of them) come first, then all the words beginning with -½, then 0, and so on. Then the words beginning with -½ are ordered using the second letter, and so on. The origin is not a root, and so does not appear in the list; but if it were, it would appear precisely half-way through the list. The roots that occur after this half-way point we denote positive roots.

Now, what is the first positive root in the list? Its 8-letter 'word' will begin with as many 0's as possible; then the first non-zero 'letter' or coordinate will be as small as possible, but positive.

The very first positive root is in fact

a = (0, 0, 0, 0, 0, 0, 0, 1).

The half-integral roots are ruled out, at this stage, because they cannot have as many 0's on the left. In fact, the first half-integral root to appear in the list will be

e = (0, 0, 0, ½, -½, 0, -½, -½).

Before that, we'll find the simple roots

b = (0, 0, 0, 0, 0, 0, 1, 0)
c = (0, 0, 0, 0, 0, 1, 0, 0)
d = (0, 0, 0, 0, 1, 0, 0, 0).

Note that the -½'s in e can be turned into +½'s by adding a, b or d. So the next root in the list that cannot be obtained as a linear combination of the simple roots already discovered, is

f = (0, 0, ½, -½, 0, -½, -½, 0).

After that we find

g = (0, ½, -½, -½, -½, 0, 0, 0)

and

h = (½, -½, -½, 0, 0, -½, 0, 0).

(Note that the pairs occuring in e, f, g, h have the patterns 0123, 0312, 1320, and 3210, respectively - all even permutations of 0123.)

We now have in our possession eight simple roots, out of which all positive roots may be formed by positive superpositions.

We now appeal to the fact that root systems can be classified by the geometrical configuration of their simple roots. We form the Cartan matrix which records the dot-products between the simple roots:

 | a  b  c  d  e  f  g  h
-------------------------
a| 1  0  0  0 -½  0  0  0
b| 0  1  0  0 -½ -½  0  0
c| 0  0  1  0  0  0 -½ -½
d| 0  0  0  1 -½  0 -½  0
e|-½ -½  0 -½  1  0  0  0
f| 0 -½ -½  0  0  1  0  0
g| 0  0  0 -½  0  0  1  0
h| 0  0 -½  0  0  0  0  1

An alternative way to encode these geometrical relationships is the Dynkin diagram. Here each simple root is represented by a dot; each non-zero dot-product by a line between two dots. Starting with e, the simple root that is most connected to other simple roots, it is straightforward to draw the diagram corresponding to the above matrix:



The connectedness of the diagram means that the simple roots cannot be split into 2 camps, belonging in mutually orthogonal subspaces of 8-dimensional space. The root system is said to be 'irreducible'. This particular diagram is the signature of the root system called E8; it gives rise to the largest of the exceptional Lie groups, also called E8.

This suffices to establish that the lattice we have obtained from the cell-centres of the 120-cell is indeed the E8 lattice. In the next post (Part 4) we will be a bit more explicit about this, and show how our lattice can be transformed into a more familiar representation of E8.