In mathematics, the McKay graph of a finite-dimensional representation V of a finite group G is a weighted quiver encoding the structure of the representation theory of G. Each node represents an irreducible representation of G. If
χ
i
,
χ
j
are irreducible representations of G then there is an arrow from
χ
i
to
χ
j
if and only if
χ
j
is a constituent of the tensor product
V
⊗
χ
i
. Then the weight nij of the arrow is the number of times this constituent appears in
V
⊗
χ
i
. For finite subgroups H of GL(2, C), the McKay graph of H is the McKay graph of the canonical representation of H.
If G has n irreducible characters, then the Cartan matrix cV of the representation V of dimension d is defined by
c
V
=
(
d
δ
i
j
−
n
i
j
)
i
j
, where δ is the Kronecker delta. A result by Steinberg states that if g is a representative of a conjugacy class of G, then the vectors
(
(
χ
i
(
g
)
)
i
are the eigenvectors of cV to the eigenvalues
d
−
χ
V
(
g
)
, where
χ
V
is the character of the representation V.
The McKay correspondence, named after John McKay, states that there is a one-to-one correspondence between the McKay graphs of the finite subgroups of SL(2, C) and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie Algebras.
Let G be a finite group, V be a representation of G and
χ
be its character. Let
{
χ
1
,
…
,
χ
d
}
be the irreducible representations of G. If
V
⊗
χ
i
=
∑
j
n
i
j
χ
j
,
then define the McKay graph
Γ
G
of G as follow:
To each irreducible representation of G corresponds a node in
Γ
G
.
There is an arrow from
χ
i
to
χ
j
if and only if nij > 0 and nij is the weight of the arrow:
χ
i
→
n
i
j
χ
j
.
If nij = nji, then we put an edge between
χ
i
and
χ
j
instead of a double arrow. Moreover, if nij = 1, then we do not write the weight of the corresponding arrow.
We can calculate the value of nij by considering the inner product. We have the following formula:
n
i
j
=
⟨
V
⊗
χ
i
,
χ
j
⟩
=
1
|
G
|
∑
g
∈
G
V
(
g
)
χ
i
(
g
)
χ
j
(
g
)
¯
,
where
⟨
⋅
,
⋅
⟩
denotes the inner product of the characters.
The McKay graph of a finite subgroup of GL(2, C) is defined to be the McKay graph of its canonical representation.
For finite subgroups of SL(2, C), the canonical representation is self-dual, so nij = nji for all i, j. Thus, the McKay graph of finite subgroups of SL(2, C) is undirected.
In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of SL(2, C) and the extended Coxeter-Dynkin diagrams of type A-D-E.
We define the Cartan matrix cV of V as follow:
c
V
=
(
d
δ
i
j
−
n
i
j
)
i
j
,
where
δ
i
j
is the Kronecker delta.
If the representation V of a finite group G is faithful, then the McKay graph of V is connected.
The McKay graph of a finite subgroup of SL(2, C) has no self-loops, that is, nii = 0 for all i.
The weights of the arrows of the McKay graph of a finite subgroup of SL(2, C) are always less or equal than one.
Suppose G = A × B, and there are canonical irreducible representations cA and cB of A and B respectively. If
χ
i
, i = 1, ..., k, are the irreducible representations of A and
ψ
j
, j = 1, ..., l, are the irreducible representations of B, then
χ
i
×
ψ
j
1
≤
i
≤
k
,
1
≤
j
≤
l
are the irreducible representations of
A
×
B
, where
χ
i
×
ψ
j
(
a
,
b
)
=
χ
i
(
a
)
ψ
j
(
b
)
,
(
a
,
b
)
∈
A
×
B
. In this case, we have
⟨
(
c
A
×
c
B
)
⊗
(
χ
i
×
ψ
l
)
,
χ
n
×
ψ
p
⟩
=
⟨
c
A
⊗
χ
k
,
χ
n
⟩
⋅
⟨
c
B
⊗
ψ
l
,
ψ
p
⟩
.
Therefore, there is an arrow in the McKay graph of G between
χ
i
×
ψ
j
and
χ
k
×
ψ
l
if and only if there is an arrow in the McKay graph of A between
χ
i
and
χ
k
and there is an arrow in the McKay graph of B between
ψ
j
and
ψ
l
. In this case, the weight on the arrow in the McKay graph of G is the product of the weights of the two corresponding arrows in the McKay graphs of A and B.
Felix Klein proved that the finite subgroups of SL(2, C) are the binary polyhedral groups. The McKay correspondence states that there is a one-to-one correspondence between the McKay graphs of these binary polyhedral groups and the extended Dynkin diagrams. For example, let
T
¯
be the binary tetrahedral group. Every finite subgroup of SL(2, C) is conjugate to a finite subgroup of SU(2, C). Consider the matrices in SU(2, C):
S
=
(
i
0
0
−
i
)
,
V
=
(
0
i
i
0
)
,
U
=
1
2
(
ϵ
ϵ
3
ϵ
ϵ
7
)
,
where ε is a primitive eighth root of unity. Then,
T
¯
is generated by S, U, V. In fact, we have
T
¯
=
{
U
k
,
S
U
k
,
V
U
k
,
S
V
U
k
|
k
=
0
,
…
,
5
}
.
The conjugacy classes of
T
¯
are the following:
C
1
=
{
U
0
=
I
}
,
C
2
=
{
U
3
=
−
I
}
,
C
3
=
{
±
S
,
±
V
,
±
S
V
}
,
C
4
=
{
U
2
,
S
U
2
,
V
U
2
,
S
V
U
2
}
,
C
5
=
{
−
U
,
S
U
,
V
U
,
S
V
U
}
,
C
6
=
{
−
U
2
,
−
S
U
2
,
−
V
U
2
,
−
S
V
U
2
}
,
C
7
=
{
U
,
−
S
U
,
−
V
U
,
−
S
V
U
}
.
The character table of
T
¯
is
Here
ω
=
e
2
π
i
/
3
. The canonical representation is represented by c. By using the inner product, we have that the McKay graph of
T
¯
is the extended Coxeter-Dynkin diagram of type
E
~
6
.
