Why ADE Dynkin diagrams classify both simply-laced Lie algebras and finite subgroups of SU(2). A direct bridge from Platonic rotations to the exceptional series.
The previous explainer closed the classification of simply-laced simple Lie algebras: An, Dn, E6, E7, E8. This looks like a statement purely about Euclidean root systems. It is not.
The same list classifies the finite subgroups of SU(2). Platonic rotations produce the same Dynkin diagrams from an entirely different starting point: their irreducible representations. This is the McKay correspondence (1980), the first hint that ADE Dynkin diagrams are a deeper combinatorial object, and the first step toward monstrous moonshine.
This explainer builds the correspondence: pick a finite subgroup G ⊂ SU(2), visualise it on S3, list its irreducible representations, tensor each with the fundamental 2-dimensional rep, draw the resulting McKay quiver. What comes out is always an affine ADE Dynkin diagram with one extra node for the trivial representation.
Two finite lists, both with the same letters and indices, proved a century apart from very different premises.
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 — single-edge trees fixed by positive-definiteness.
The finite-subgroup classification of SU(2) (Klein 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 double covers of the Platonic rotation groups.
McKay's bridge: if G ⊂ SU(2) is finite, the McKay quiver of G equals the affine Dynkin diagram Ã, D̃, Ẽ6,7,8 of the corresponding type. The two classifications share an explicit combinatorial object.
Affine versus finite. The Lie-algebra classification gives finite diagrams with rank nodes; the McKay quiver is the affine version with one extra node (the trivial representation). Deleting the affine node recovers the finite diagram.
SU(2) is the unit 3-sphere: writing elements as [[a, b], [-b̄, ā]] with |a|² + |b|² = 1 gives S3 ⊂ ℂ2. Every finite subgroup is a finite point configuration on S3, closed under quaternion multiplication.
The five subgroup types:
n points equispaced on a great circle of S3.Project S3 stereographically to ℝ3 and render the resulting point cloud; the next figure lets you pick a group and rotate.
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, 2I sit in highly symmetric configurations — 2I in particular is the set of 120 unit icosians, a famously rigid discrete object in ℍ.
Given a finite subgroup G ⊂ SU(2), the construction has four steps.
Step 1. List the irreducible complex representations of G. Call them ρ0 (the trivial representation, always present), ρ1, …, ρn. There are finitely many; in fact the number of irreps equals the number of conjugacy classes of G.
Step 2. Take V = ℂ2, the standard 2-dimensional representation of SU(2) restricted to G. This is the fundamental representation.
Step 3. For each irrep ρi, form the tensor product V ⊗ ρi. This is another representation of G, generally reducible. Decompose it:
where the integers aij ≥ 0 count how many copies of ρj appear in V ⊗ ρi. A remarkable fact: for a finite subgroup G ⊂ SU(2), every aij ∈ {0, 1}.
Step 4. Build a graph, the McKay quiver: one vertex per irrep ρi, and an edge between ρi and ρj whenever aij = 1. Because V is self-dual (SU(2) has V ≅ V*), the matrix aij is symmetric and we get an undirected graph.
This graph is the affine Dynkin diagram for G. The affine node is always ρ0, the trivial representation.
Each group has rank(ADE) + 1 irreducible representations (the affine diagram has one more node than the finite one). The character tables are classical.
By Burnside, the sum of squares of irrep dimensions equals |G|. For 2I: 1² + 2² + 2² + 3² + 3² + 4² + 4² + 5² + 6² = 120.
Figure 2. Bar chart of irrep dimensions for the selected subgroup of SU(2). The sum of squares (Σ dim² = |G|) is Burnside's identity for finite groups. The number of irreps equals the affine Dynkin diagram's node count, which is rank + 1 for the corresponding ADE Lie algebra.
Each irrep ρi becomes a node. The edges come from decomposing V ⊗ ρi. The figure below animates this one step at a time: pick a group, then step through each irrep ρ0, ρ1, … and watch the edges appear as its tensor decomposition is computed.
For ℤ/n, irreps are 1-dimensional characters χk: g ↦ ζk, ζ = e2πi/n. V = χ1 ⊕ χ-1, so V ⊗ χk = χk+1 ⊕ χk-1: each node connects to its two neighbours mod n, the affine Ãn-1 diagram.
For the binary polyhedral groups, decomposition matrices are tabulated in standard references (Humphreys, Slodowy).
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 an edge from ρi to every ρj with aij = 1. The final graph is the affine Dynkin diagram of the corresponding ADE algebra.
Strip the affine node ρ0 from the McKay quiver and you recover the finite Dynkin diagram that classifies the corresponding simply-laced simple Lie algebra. The figure below puts the two diagrams side by side to confirm the match.
Figure 4. Left: the McKay quiver computed from the irrep decomposition. Right: the affine Dynkin diagram from the Lie-algebra classification. The two are identical as labelled graphs — the red node on the right is the affine node, which on the left is the trivial representation ρ0.
The entire correspondence fits in a single table. Click any row to synchronise all four scenes above to that choice.
| 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.
ADE Dynkin diagrams show up in simple surface singularities (the Du Val classification), cluster algebras, quivers with potentials, del Pezzo surfaces, Cartan matrices of finite reflection groups, and the sporadic groups feeding moonshine. The same five types appear wherever a finite classification meets a positively curved combinatorial structure.
McKay's 1978 observation that j-function q-expansion coefficients are sums of Monster irrep dimensions launched monstrous moonshine and led to vertex operator algebras and generalised Kac–Moody algebras. The mental move was the same as here: look at irreps of a mysterious group and see what combinatorial object they assemble. The Moonshine story is picked up in the Modular Forms series.
A physicist meeting E8 through its 248-dimensional adjoint and a geometer meeting it through the 120-element binary icosahedral group are looking at the same diagram from different directions.