THE MCKAY CORRESPONDENCEYI SUN

Abstract. The McKay correspondence gives a bijection between the finite subgroups of SU (2) and theaffine simply laced Dynkin diagrams. In particular, this bijection associates naturally to each finite dimensional representation of SU (2) a vertex of the corresponding diagram. The goal of this talk will be toconstruct this correspondence and to discuss some proofs and generalizations.

1. Introduction
Let G be a finite group and V a faithful representation of G over C, that is, an injective map G → GL(V ).
Knowledge of such a representation gives G as a group of matrices. Further, it is a classical fact that all
irreducible representations of G are contained in V ⊗N for some N . In some sense, the size of V is a measureof the complexity of G.
One might therefore be led to consider which groups G have faithful representations in rank n, especiallywhen n is small. But recall that every finite dimensional representation of a finite group may be given aunitary structure, meaning that it suffices to consider finite subgroups of U (n). Recall now that centralelements of U (n) are given by the diagonal embedding U (1) ,→ U (n) and that the quotient U (n)/U (1) bythis center is SU (n). Therefore, if we restrict our attention to simple groups, it is enough to consider finite
subgroups of SU (n).
The purpose of this talk is to present an especially elegant classification of these subgroups for the casen = 2. Of course, to obtain the classification, it is possible to simply consider the standard double coverπ : SU (2)  SO(3)and note that the only element of even order in SU (2) is the generator −1 of the kernel of π. Therefore, anyfinite subgroup of SU (2) either has even order and is the preimage of a finite subgroup of SO(3) or has oddorder and is isomorphic to a finite subgroup of SO(3) of odd order, hence a cyclic group. From this and theclassical classification of finite subgroups of SO(3) as symmetry groups of regular polyhedra, we obtain thefollowing.
Proposition 1. Any finite subgroup of SU (2) is one of the following groups:• The cyclic group Z/nZ for n > 1.
• The binary dihedral group BD2n for n > 1, the preimage of the dihedral group D2n under π.
• The binary tetrahedral group BT, the preimage of the tetrahedral group T under π.
• The binary octahedral group BO, the preimage of the octahedral group O under π.
• The binary dodecahedral group BD, the preimage of the dodecahedral group D under π.
2. Statement of the correspondence
Let G be a finite subgroup of SU (2), and let V be the faithful representation of G obtained from theembedding G ,→ SU (2). Let {Vi } be the irreducible representations of G. Define the McKay quiver of Gto be the directed multi-graph with vertices Vi and mij edges from Vi to Vj if Vj occurs mij times in thedecomposition of Vi ⊗ V into irreducibles. In other words, we haveM ⊕mVi ⊗ V =Vj ij .
jThe McKay correspondence classifies the possible groups G via their McKay quivers. More precisely, wehave the following.
Date: December 1, 2010.
Notes for a talk given on December 1, 2010 as part of the 2010 Part III Seminar Series at the University of Cambridge.
12YI SUN

Theorem 1 (McKay 1980). Let G be a non-trivial finite subgroup of SU (2). Then, the McKay quiver ofSU (2) is an affine simply laced Dynkin diagram.
Before sketching the proof, we exhibit each case of this correspondence explicitly in Table 1.
Table 1. The explicit form of the McKay correspondence.
Finite subgroup of SU (2)Affine simply laced Dynkin diagram
11Z/nZhx | xn = 1i

1

11

1

1

132

1432

en−1
A
1
1222

12BD2ne n−2hx, y, z | x2 = y 2 = y n = xyzi D

1121BThx, y, z | x2 = y 3 = z 3 = xyzi

2213e6E

21BOhx, y, z | x2 = y 3 = z 4 = xyzi

234e7E
32BD

235hx, y, z | x = y = z = xyzi

45

16e8E

3. Proof of the correspondence
We give here a proof intended to minimize the number of necessary prerequisites, but we note that
there are other “deeper” proofs possible. Of course, it is also possible to give a proof based on case-by-caseverification (which is how this correspondence was first discovered), but we would like to give a more uniform
interpretation.
Proof of Theorem 1. We will slowly obtain more and more combinatorial properties of the McKay quiver G
until the affine simply laced Dynkin diagrams pop out as the only graphs satisfying these properties.
Claim 1: The McKay quiver of any G is an undirected graph, that is, mij = mji . For this, let χi be thecharacter of the representation Vi , and notice that1 X
1 X
mij = hχi , χV χj i =χi (g)χV (g)χj (g) =χi (g)χV (g)χj (g) = hχi χV , χj i = mji ,|G||G|g∈Gg∈G

where we note that χV is real because each element of SU (2) has real trace.
Claim 2: The McKay quiver is connected. This follows from the fact that every irreducible representationof G is contained in some tensor power of the faithful representation V .
Claim 3: The McKay quiver has no self-loops, that is, mii = 0. For this, observe that
1 X
mii = hχV χi , χi i =χV (g)|χi (g)|2 .
|G|g∈GIf G has even order, then it contains the element −1 ∈ SU (2), so multiplication by −1 defines an involution
on G with χV (g) = −χV (−g); hence we have
1 X
2mii =
χV (g)(|χi (g)|2 − |χi (−g)|2 ) = 0.
|G|g∈G

THE MCKAY CORRESPONDENCE

3Otherwise, if G has odd order, then its image in SO(3) under π has odd order, hence G is a cyclic group Z/nZ
for some n. Let A ∈ SU (2) generate G, where A 6= 1 satisfies An = 1, that is, (1+A+A2 +· · ·+An−1 )(1−A) =0. But det(1 − A) 6= 0 if A 6= 1 because A ∈ SU (2), so we see that
1 X
1 X
1mii =
χV (g)|χi (g)|2 =χV (g) =Tr(1 + A + · · · + An−1 ) = 0.
|G||G||G|g∈Gg∈G

Claim 4: The McKay quiver is a simple graph, that is, mij ∈ {0, 1}. For this, notice thatsXsX
1 X
122mij =χi (g)χV (g)χj (g) ≤|χi (g)| |χV (g)||χj (g)|2|G||G|g∈Gg∈Gg∈Gs XsX
1<4|χi (g)|2|χj (g)|2 = 2.
|G|g∈Gg∈GRemainder of proof: These four properties along with the definition of the McKay quiver will be enough for
us to characterize them as the affine simply laced Dynkin
diagrams. Let di = dim Vi be the dimension of ViPand label vertex Vi by di . Then, notice that 2di = j∼i dj , where j ∼ i means i and j are neighbors.
en−1 .
Step 1: If the McKay quiver contains a cycle, then it is of type A
If i1 , . . . , ik are vertices in a cycle, then we have 2dik ≥ dik−1 +dik+1 , so summing these shows we must haveequality, meaning there are no other vertices in the graph. Further, the trivial representation has dimension
1, meaning that all di are equal to 1.
e 4.
Step 2: Each vertex of the McKay quiver has degree at most 4, and if such a vertex exists, it is of type D
Let j1 , . . . , j5 be neighbors of i. Then, we have that
15di ≤ [dj1 + · · · + dj5 ] ≤ di ,42
a contradiction, where the first inequality holds because djk ≥ 21 di . So vertices have degree at most 4. In
this case, for neighbors j1 , . . . , j4 of i, we have
1di ≤ [dj1 + · · · + dj4 ] ≤ di ,2e 4.
so equality holds in all cases, which shows that we are in type D
Step 3: The McKay quiver has a vertex of degree at least 3 connected to a leaf with label 1.
If all vertices have degree 2 or less, then the quiver is a path, which is impossible. So there is at least one
vertex of degree 3. It remains to show that there is a leaf with label 1; but if not then all vertices in thequiver will have label greater than 1, which is impossible.
fn or it is the only vertex of degree 3.
Step 4: If this vertex of degree 3 has label 2, then we are in type D
Let i and j be two vertices with degree 3 connected by a path, let x, y be the two neighbors of i not on
this path, and let z, w be the two neighbors of j not on this path. Then, we have by the conditions on thepath and the fact that 2dx , 2dy ≥ di and 2dz , 2dz ≥ dj thatdi + dj ≥ dx + dy + dz + dw ≥ di + dj ,so we have equality. Therefore, there are no other vertices abutting the path and x, y, z, w are the onlye n.
neighbors of i and j. This is D
f6 , Ef7 , or E
f8 .
Step 5: If there is only one vertex of degree 3, then we are in type E
This is clear by easy computation.
4. Other interpretations and generalizations
We have presented the McKay correspondence as only between finite subgroups of SU (2) and ADE-typeaffine Dynkin diagrams. However, it extends also to a different context, which we will only sketch here.
Finite subgroups G of SU (2) ↔ simply laced affine Dynkin diagrams ↔ Resolution graphs of C2 /GBecause G ⊂ SU (2), it acts on C2 and we may consider the quotient variety C2 /G, which has a singularity
at 0. In this case, there is a minimal resolution of C2 /G, i.e. smooth variety π : X  C2 /G where the4YI SUN

preimage π −1 (0) of 0 ∈ C2 /G is the union of irreducible components isomorphic to CP1 . Then, the graphwith these components as vertices and an edge between two vertices if their components intersect is theresolution graph of C2 /G. It turns out that this graph is just the simply laced affine Dynkin diagram ofG! Further, the K-theory K0 (X) of X is simply the representation ring of G! And even further, thisisomorphism extends to an equivalence between the derived categories Db (Coh(C2 )G ) and Db (Coh(X)) ofG-equivariant coherent sheaves on C2 and coherent sheaves on X!
Sources
The correspondence was first noticed by J. McKay in [4]. The proof we give here is based loosely on one
suggested in [1] together with an exercise assigned by A. Postnikov (Problem 5 in [5]). The connection tothe K-theory of Kleinian singularities was given in [2], and its extension to a derived equivalence in [3].
References[1] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina, and D. Vaintrob, Introduction to Representation
Theory, available at: http://arxiv.org/abs/0901.0827.
[2] G. Gonzalez-Sprinberg and J. L. Verdier, Construction géométrique de la correspondance de McKay, Ann. ENS 16 (1983),
409-449.
[3] M. Kapranov, E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), 565-576.
[4] J. McKay, Graphs, singularities, and finite groups, The Santa Cruz Conference on Finite Groups (Univ. California, SantaCruz, California, 1979), Proc. Sympos. Pure Math 37, American Mathematical Society, Providence, RI 1980, 183-186.
[5] A. Postnikov, Problem Set #2, 18.319 Spring 2008, available at: http://math.mit.edu/~apost/courses/18.319/pset2.
pdf.