Part 2 of 9. Every root carries a copy of sl2(ℂ), whose representation theory forces an integer constraint on every pair of roots. Reflections in the roots then generate a finite group, the Weyl group.
Part 1 put a Euclidean inner product on the span of the roots but said nothing about how the roots can sit. The constraint comes from a copy of sl2(ℂ) attached to each root, whose Cartan element can only have integer eigenvalues. Each root α then gives a reflection sα of 𝔼 that permutes Δ, and these reflections generate the Weyl group W.
Fix α ∈ Δ. By Part 1, 𝔤α and 𝔤−α are one-dimensional and [𝔤α, 𝔤−α] ⊆ 𝔥. Nonzero eα ∈ 𝔤α and fα ∈ 𝔤−α, suitably rescaled, satisfy the sl2 relations with hα = [eα, fα]:
So 𝔤α ⊕ 𝔤−α ⊕ ℂhα is a copy of sl2(ℂ), one for each pair {α, −α}, and facts about sl2 transport to 𝔤.
One fact about sl2(ℂ) is used without proof: in every finite-dimensional representation, H acts diagonally with integer eigenvalues that are symmetric about 0 (if k occurs, so does −k, with the same multiplicity).
The adjoint action makes 𝔤 a representation of the copy of sl2 at α, and hα acts on 𝔤β by the number β(hα). So that number is an integer, the Cartan integer
The roots β + kα form an unbroken α-string β − pα, …, β + qα, and the sum of their root spaces is one irreducible representation, with eigenvalues nβ,α + 2k for k from −p to q. Symmetry about 0 forces p − q = nβ,α.
Figure 1. α-strings. Click a root to make it β; the string through it runs parallel to α (amber), and hollow marks are the first non-roots at each end. Right: hα-eigenvalues along the string, always symmetric about 0. Bottom: string lengths over all ordered pairs β ≠ ±α, by type.
Define sα: 𝔼 → 𝔼 by
It is the reflection across the hyperplane orthogonal to α: sα(α) = −α, and sα fixes every vector perpendicular to α. On a root β it reverses the α-string through β, so sα(β) is again a root:
The Weyl group is the group these reflections generate,
W permutes Δ, and since Δ spans 𝔼 an element of W is determined by that permutation, so W is finite.
Reflections preserve length, so an orbit of W never mixes root lengths. In an irreducible root system each length is a single orbit: one orbit when all roots have the same length (the simply-laced types A, D and E), two otherwise (B, C, F4, G2).
Figure 2. Weyl orbit explorer. Click a root to highlight its orbit. In A2 every click lights the whole hexagon. B2 splits into the four long and the four short roots; G2 into six short and six long, with length ratio √3. Dashed grey lines are the mirrors of the simple reflections.
Since nα,β nβ,α = 4 cos2 θ, which is less than 4 for roots that are not proportional, their Cartan integers lie in {0, ±1, ±2, ±3}, and a pair of unequal values, like −2 and −1 for the simple roots of B2, means unequal lengths. Part 3 turns this into the four possible angles between roots.
In E8 all 240 roots have squared length 2, so sα(x) = x − (x, α) α. A simple reflection fixes the 126 roots perpendicular to αi and moves the other 114. Figure 3 draws each root as a polyline across eight parallel axes, one per coordinate.
Figure 3. One simple reflection splits the 240 roots of E8 into the fixed (grey) and the moved (amber). A root is fixed exactly when it is perpendicular to the chosen αi, and for every i that is 126 roots.
Because E8 is irreducible and simply-laced, the orbit of α1 is all 240 roots.
Figure 4. The orbit of α1 in E8. Each click applies a simple reflection to every root collected so far and keeps the new ones. In any order, the eight reflections end at 240 roots.
Orbits give the order of W. The stabiliser of a fundamental weight ωi (the vector with 2(ωi, αj)/(αj, αj) = 1 for j = i and 0 otherwise) is the Weyl group of the diagram with node i deleted, so |W| = |W · ωi| · |W(diagram − i)|. Repeating the step until no nodes remain writes |W| as a product of orbit sizes. In E8, ω8 is the highest root, so its orbit is the 240 roots and its stabiliser is W(E7).
Figure 5. |W| by orbit and stabiliser. Each row deletes one node (amber) and counts, by applying reflections until nothing new appears, the orbit of its fundamental weight under the nodes still present. Click another node to take a different route; the product does not change.
For E8 that is 696,729,600 = 214 · 35 · 52 · 7, which is 1,920 times |W(A8)| = 9! and 135 times |W(D8)| = 27 · 8!, though all three act on eight dimensions.
The length of an element of W is the fewest simple reflections whose product it is. A breadth-first search from the identity reaches each element at its length, and the longest length is always the number of positive roots, held by exactly one element. For E8 the search would visit 696,729,600 elements, so Figure 6 also reads the distribution off the heights of the positive roots (a theorem of Kostant) and checks that reading against the search wherever the search runs.
Figure 6. Elements of W counted by length. Bars come from breadth-first search from the identity (outlined bars, for E7 and E8, from root heights alone); dots are the counts read off the heights of the positive roots. The counts are symmetric and stop at the number of positive roots.