From the 120-cell to E8 (Part 4)

A more explicit definition of the E8 lattice runs as follows. The point with coordinates (x1,x2,...x8) belongs to the lattice iff

either x1..x8 are all integers, and their sum is even
or x1..x8 are all half-integral (i.e. an integer plus ½) and their sum is an even integer

In this case, the lattice-points closest to the origin, such as

(½, ½, -½, ½, ½, -½, ½, ½)

or

(0, 1, 0, 0, 0, -1, 0, 0)

are all at distance √2. But once again, there are 240 of them (128 of the half-integral kind, 28×4=112 of the integral kind). And once again we can arrange them in 'alphabetical order', and find a set of simple roots.

The first positive root is

A = (0, 0, 0, 0, 0, 0, 1,-1);

the second (linearly independent of the first) is

B = (0, 0, 0, 0, 0, 0, 1, 1).

After that we have a run of five:

C = (0, 0, 0, 0, 0, 1,-1, 0);
D = (0, 0, 0, 0, 1,-1, 0, 0);
E = (0, 0, 0, 1,-1, 0, 0, 0);
F = (0, 0, 1,-1, 0, 0, 0, 0);
G = (0, 1,-1, 0, 0, 0, 0, 0);

and finally

H = (½, -½, -½, -½, -½, -½, -½, ½)

(note: the number of minus signs must be even). The Dynkin diagram is


Recalling the diagram generated by our first lattice,


we see that the simple roots correspond as follows:

a → B
b → D
c → F
d → A
e → C
f → E
g → H
h → G.

This correspondence of the simple roots determines a linear transformation from the first space to the second, as follows. The coordinate basis vectors e5 ( = (0, 0, 0, 0, 1, 0, 0, 0)), e6, e7, e8 are none other than d, c, b, a and map to A, F, D, B, respectively. Thus

e5 maps to A = (0, 0, 0, 0, 0, 0, 1,-1);
e6 maps to F = (0, 0, 1,-1, 0, 0, 0, 0);
e7 maps to D = (0, 0, 0, 0, 1,-1, 0, 0);
e8 maps to B = (0, 0, 0, 0, 0, 0, 1, 1);

e4 is 2e + a + b + d and maps to 2C + B + D + A = (0, 0, 0, 0, 1, 1, 0, 0);

e3 is 2f + e4 + b + c and maps to 2E + 3C + 2B + D + A = (0, 0, 1, 1, 0, 0, 0, 0);

e2 is 2g + e3 + e4 + d and maps to 2H + (0, 0, 1, 1, 1, 1, 0, 0) + A
= (1, -1, 0, 0, 0, 0, 0, 0);

e1 is 2h + e2 + e3 + c and maps to 2G + (1, -1, 1, 1, 0, 0, 0, 0) + F
= (1, 1, 0, 0, 0, 0, 0, 0).

So the matrix for mapping a vector in the first space to a vector in the second (both written as column vectors) takes the remarkably simple form:

[ 1  1  0  0  0  0  0  0  ]
[ 1 -1  0  0  0  0  0  0  ]
[ 0  0  1  0  0  1  0  0  ]
[ 0  0  1  0  0 -1  0  0  ]
[ 0  0  0  1  0  0  1  0  ]
[ 0  0  0  1  0  0 -1  0  ]
[ 0  0  0  0  1  0  0  1  ]
[ 0  0  0  0 -1  0  0  1  ]

The transformation could be expressed verbally as follows. First permute the coordinates of your vector, from

(x1,x2,x3,x4,x5,x6,x7,x8)

to

(x1,x2,x3,x6,x4,x7,x8,x5),

then take the coordinates in pairs and act on them with the matrix

[ 1  1 ]
[ 1 -1 ].

This last step takes the pair

(0,0) (which we have called 0) to (0,0)
(0,½) or 1 to (½,-½)
(½,0) or 2 to (½,½) and
(½,½) or 3 to (1,0).

To arrive at a vector of the desired type (i.e. a root of the E8 system in the conventional coordinates), the product of the first stage of the transformation must either be a mixture of the pairs 0 and 3, or else a mixture of 1 and 2 (giving rise to integral vectors and half-integral vectors, respectively).

Further, for the sum of the ccordinates to be even, the number of occurrences of 3 (in the integral case) or 2 (in the half-integral case) must be even, that is, 0, 2 or 4. The possibilities are

0033, 1111, 1122, 2222 (and permutations thereof).

These are 6+1+6+1=14 in number. Multiplying by 16, for all the sign combinations, that is 224 possibilities. Once more there is an opportunity for mathematical pottering. It is amusing to verify that the even permutations of 0123, together with 1111 and 2222, are transformed by the first stage (permutation of the last five coordinates) into patterns of the required form. For example

0123 = (0, 0, 0, 1, 1, 0, 1, 1) is permuted into (0, 0, 0, 0, 1, 1, 1, 1) = 0033,

and so on. Odd permutations of the pairs give rise, of course, to sequences that are illegal in one way or another.

This concludes our exploration of the way that leads from the 120-cell to E8. It has taken quite a bit of explaining, but with hindsight we can see that it is really only a little bit of a tweak that turns one into the other.

I recommend David Richter's web site for more on the 600-cell and the E8 root system (especially this page). The splendid photograph on the front page shows Richter holding a model which is a projection into 3 dimensions of the 600-cell. The sun shines through the model and casts a shadow on the ground. The beauty of the shadow is that it corresponds to something known as the Van Oss projection of the 600-cell. It bears a more than passing resemblance to a well-known diagram of the E8 root-system (see for example here).