Part 5 of 9. The ADE diagrams that classify simply-laced Lie algebras also classify the finite subgroups of SU(2). McKay's construction reads each diagram directly off a group's representations.
Part 4 finished the list of simply-laced simple Lie algebras: An, Dn, E6, E7, E8. The same list classifies the finite subgroups of SU(2). This is the McKay correspondence (1980).
Klein (1884) showed that every finite G ⊂ SU(2) is cyclic ℤ/n, binary dihedral BD4n, or a double cover of a Platonic rotation group: binary tetrahedral 2T, octahedral 2O or icosahedral 2I. McKay turned each group into a graph, the McKay quiver, and found the affine Dynkin diagram of the matching type: Ã for cyclic groups, D̃ for binary dihedral ones, Ẽ6, Ẽ7, Ẽ8 for 2T, 2O, 2I. The extra affine node is the trivial representation; deleting it leaves the finite diagram.
Writing elements of SU(2) as [[a, b], [-b̄, ā]] with |a|² + |b|² = 1 identifies SU(2) with the unit quaternions, the 3-sphere S3. A finite subgroup is a finite set of points on S3 closed under multiplication:
Dots: elements of G, projected stereographically from S3 to ℝ3 and radially compressed. The sphere is the equator q₀ = 0; elements with q₀ > 0 lie inside it. The identity sits at the centre; −1 goes to infinity and is not drawn.
Figure 1. Finite subgroups of SU(2) as points on the 3-sphere. The cyclic group ℤ/n sits on one great circle; 2T, 2O and 2I spread over the whole sphere. Drag to rotate.
List the irreducible complex representations of G: ρ0 (the trivial one), ρ1, …, one for each conjugacy class. Let V = ℂ2 be the defining representation of SU(2) restricted to G, and decompose each tensor product V ⊗ ρi:
The McKay quiver has one node per irrep and aij edges from ρi to ρj; V is self-dual, so aij = aji. Each aij is 0 or 1 except for ℤ/2, whose two nodes are joined by the double edge of Ã1, and the trivial group, a single node with a double loop.
All of this can be read from the character table, the trace of each group element in each irrep: V ⊗ ρi has character χVχi, and aij is its inner product with χj, an average over the group.
Figure 2. The character table, computed from the group's quaternions: one column per conjugacy class, one row per irreducible representation. Click a row ρi to add the row χVχi below; the highlighted rows sum to it, column by column, giving the aij.
The identity column lists the dimensions, and its squared length is |G|. The row of V is 2 Re g, twice the real part of each quaternion.
Figure 3. The first column of Figure 2 as bars: the dimension of each irreducible representation. The squares sum to |G|, and the number of bars is the node count of the affine diagram.
The cyclic case can be done by hand. The irreps of ℤ/n are χk(g) = ζk with ζ = e2πi/n, and V = χ1 ⊕ χ−1, so V ⊗ χk = χk+1 ⊕ χk−1: the graph is the cycle Ãn−1. Figure 4 does the same for any group.
Black node = trivial rep ρ0 (the affine node). Coloured nodes = other irreps, labelled by dimension. Each edge encodes a summand of V ⊗ ρi.
Figure 4. Building the McKay quiver one irrep at a time. Each step decomposes V ⊗ ρi using the characters of Figure 2 and draws an edge to every constituent.
Figure 5 sets the result beside the affine Dynkin diagram from the Lie-algebra side.
Figure 5. Left, the quiver computed from the characters; right, the affine Dynkin diagram drawn from the classification. The page searches for a graph isomorphism that sends ρ0 to the affine node (red) and reports whether it found one.
Evaluate V ⊗ ρi = ⊕ aij ρj at the identity: 2di = Σj aij dj, with di = dim ρi. So the dimension vector d is an eigenvector of the adjacency matrix A with eigenvalue exactly 2, and (2I − A)d = 0. Since 2I − A is the affine Cartan matrix, d is the null vector that put affine diagrams on Part 4's boundary: the dimensions are the marks. For 2I they run 1, 2, 3, 4, 5, 6 along the long arm from ρ0, then 4, 2 and 3 on the other two.
Evaluating at another class g gives another eigenvector, that column of the character table, with eigenvalue χV(g) = 2 Re g. Delete ρ0 and the top eigenvalue drops below 2.
Figure 6. Eigenvalues λ = 2 cos θ of each McKay graph, placed by θ. Filled: the affine graph, at the class rotation angles, always including λ = 2. Hollow: ρ0 deleted, at multiples of π/h, h the sum of the dimensions. Click a row to select it.
The finite eigenvalues land on 2 cos(πm/h), with h = Σ di the Coxeter number: 12, 18 and 30 for E6, E7 and E8. The 30 is the one behind the Coxeter plane of Part 8.
McKay's 1978 observation that the j-function's coefficients are sums of Monster irrep dimensions, which started monstrous moonshine, also reads numbers off a group's irreps; see the Modular Forms series. Part 6 doubles the quaternions once more, to the octonions.