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 also classifies the finite subgroups of SU(2): the cyclic and binary dihedral groups give An and Dn, the double covers of the Platonic rotation groups give E6, E7 and E8, and each group produces its diagram through its irreducible representations. This is the McKay correspondence (1980). This part asks how a group's representations produce a Dynkin diagram.
The construction has five steps: pick a finite subgroup G ⊂ SU(2), draw it on S3, list its irreducible representations, tensor each with the 2-dimensional representation, and draw the resulting McKay quiver. The quiver is always an affine ADE Dynkin diagram, with the extra node belonging to the trivial representation.
The Lie-algebra classification (Killing, Cartan, Dynkin) says every simply-laced simple complex Lie algebra has Dynkin diagram An, Dn (n ≥ 4), E6, E7 or E8, the single-edge trees allowed by positive-definiteness.
Klein's classification (1884) says every finite G ⊂ SU(2) is cyclic ℤ/n, binary dihedral BD4n, binary tetrahedral 2T, binary octahedral 2O or binary icosahedral 2I. The last three are the double covers of the Platonic rotation groups.
The McKay correspondence says that the McKay quiver of a finite G ⊂ SU(2) is the affine Dynkin diagram Ã, D̃ or Ẽ6,7,8 of the matching type. The affine diagram has one node more than the finite one, and deleting that node (here, the trivial representation) recovers 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 3-sphere S3 ⊂ ℂ2, or equivalently with the unit quaternions. A finite subgroup is a finite set of points on S3 closed under quaternion multiplication:
n points equispaced on a great circle.Figure 1 projects S3 stereographically to ℝ3 so you can pick a group and rotate it.
Coloured dots: elements of G ⊂ SU(2) ≅ S3, stereographically projected from S3 to ℝ3. The translucent sphere is the equator |q₀| = 0; opposite poles of S3 project to the origin and infinity.
Figure 1. Finite subgroups of SU(2) as point configurations on the 3-sphere. The cyclic group ℤ/n sits on a single great circle; the binary polyhedral groups 2T, 2O and 2I sit in far more symmetric arrangements.
First, list the irreducible complex representations of G: ρ0 (the trivial one), ρ1, …, ρn. There is one for each conjugacy class of G. Second, take V = ℂ2, the defining representation of SU(2) restricted to G. Third, for each ρi decompose the tensor product V ⊗ ρi, which is usually reducible:
where the integers aij ≥ 0 count how many copies of ρj appear in V ⊗ ρi. For every finite subgroup of SU(2) except two, each aij is 0 or 1. One exception is ℤ/2 = {±1}: there −1 acts on V as −I, so V is two copies of the sign character, V ⊗ ρ0 = 2ρ1, and the graph has a double edge. That matches affine Ã1, whose Cartan matrix has off-diagonal entries −2. The other is the trivial group, where V is two copies of ρ0, so a00 = 2 and the graph is a single node with a double loop, the degenerate Ã0.
Finally, draw the McKay quiver: one vertex per irrep, and aij edges between ρi and ρj. V is self-dual (V ≅ V* for SU(2)), so aij is symmetric and the graph is undirected. It is the affine Dynkin diagram for G, with ρ0 as the affine node.
Each group has rank + 1 irreducible representations, one per node of the affine diagram. The sum of the squares of their dimensions is |G|; for 2I, 1² + 2² + 2² + 3² + 3² + 4² + 4² + 5² + 6² = 120.
Figure 2. Irrep dimensions for the selected subgroup. The squares sum to |G|, and the number of bars is the node count of the affine Dynkin diagram.
The cyclic case can be done by hand. The irreps of ℤ/n are the characters χk: g ↦ ζk with ζ = e2πi/n. V = χ1 ⊕ χ-1, so V ⊗ χk = χk+1 ⊕ χk-1, and each node joins its two neighbours mod n. That is the cycle Ãn-1. For the binary polyhedral groups the decompositions are tabulated (Humphreys, Slodowy), and Figure 3 computes them one irrep at a time.
Black node = trivial rep ρ0 (the affine node). Coloured nodes = other irreps, labelled by dimension. Each edge encodes a summand of V ⊗ ρi.
Figure 3. Building the McKay quiver step by step. For each irrep ρi, compute V ⊗ ρi = ⊕ aij ρj and draw aij edges from ρi to ρj. Everything is computed in the page: the group is generated from its quaternions, the irreducible characters are found by tensoring with V and splitting off new constituents, and each aij is an inner product of characters.
Figure 4 sets the computed quiver beside the affine Dynkin diagram from the Lie-algebra side.
Figure 4. Left: the McKay quiver computed from the irrep decomposition. Right: the affine Dynkin diagram from the Lie-algebra classification. The page searches for an isomorphism between the two graphs that sends the trivial representation ρ0 to the affine node (red), and reports whether it found one.
Figure 5 lists the families side by side.
| Finite subgroup G ⊂ SU(2) | |G| | # irreps | ADE type | Affine Dynkin |
|---|
Figure 5. The five McKay families (cyclic, binary dihedral, 2T, 2O, 2I), with the binary dihedral row sampled at n = 2 and n = 3 to show the D̃ pattern. Click a row to switch the other figures. The infinite à family is represented here by ℤ/5 (= Ã4); D̃4 (BD8, the quaternion group Q8) is the smallest non-trivial binary-dihedral case.
The dimension of each irreducible representation equals the mark of its node, the coefficient of that simple root in the highest root, with the affine node marked 1. For 2I the nine dimensions are 1, 2, 3, 4, 5, 6, 4, 3, 2, and the marks of Ẽ8 are 1 on the affine node and 2, 3, 4, 6, 5, 4, 3, 2 on the others (Part 4); under the isomorphism of Figure 4 they agree node for node. Deleting ρ0 recovers the finite E8 diagram.
ADE diagrams turn up elsewhere too: in simple surface singularities (Du Val), cluster algebras and del Pezzo surfaces. McKay's 1978 observation that the coefficients of the j-function are sums of Monster irrep dimensions made the same move as here, reading a combinatorial object off the irreps of a group, and it started monstrous moonshine. That story is in the Modular Forms series.
The groups in this part are sets of unit quaternions. Part 6 doubles the quaternions once more, to the octonions.