McKay graph

Affine (extended) Dynkin diagrams

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 $V \otimes χ_{i} .$ Then the weight $n ij$ of the arrow is the number of times this constituent appears in $V \otimes χ_{i} .$ For finite subgroups $H$ of ⁠ $GL (2, ℂ),$ ⁠ the McKay graph of $H$ is the McKay graph of the defining 2-dimensional representation of $H$ .

If $G$ has $n$ irreducible characters, then the Cartan matrix $c V$ 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 Robert 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 $c V$ 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.

Definition

Let $G$ be a finite group, $V$ be a representation of $G$ and $χ$ be its character. Let ${χ_{1}, \dots, χ_{d}}$ be the irreducible representations of $G$ . If

V \otimes χ_{i} = \sum_{j} n_{i j} χ_{j},

then define the McKay graph $Γ G$ of $G$ , relative to $V$ , as follows:

Each irreducible representation of $G$ corresponds to a node in $Γ G$ .
If $n ij > 0$ , there is an arrow from $χ i$ to $χ j$ of weight $n ij$ , written as $χ_{i} \overset{n_{i j}}{\to} χ_{j},$ or sometimes as $n ij$ unlabeled arrows.
If $n_{i j} = n_{j i},$ we denote the two opposite arrows between $χ i, χ j$ as an undirected edge of weight $n ij$ . Moreover, if $n_{i j} = 1,$ we omit the weight label.

We can calculate the value of $n ij$ using inner product $⟨ \cdot, \cdot ⟩$ on characters:

n_{i j} = ⟨ V \otimes χ_{i}, χ_{j} ⟩ = \frac{1}{| G |} \sum_{g \in G} V (g) χ_{i} (g) \overline{χ_{j} (g)} .

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 on ⁠ $ℂ^{2}$ ⁠ is self-dual, so $n_{i j} = n_{j i}$ 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 $c V$ of $V$ as follows:

c_{V} = (d δ_{i j} - n_{i j})_{i j},

where $δ ij$ is the Kronecker delta.

Some results

If the representation $V$ is faithful, then every irreducible representation is contained in some tensor power $V^{\otimes k},$ and the McKay graph of $V$ is connected.
The McKay graph of a finite subgroup of ⁠ $SL (2, ℂ)$ ⁠ has no self-loops, that is, $n_{i i} = 0$ for all $i$ .
The arrows of the McKay graph of a finite subgroup of ⁠ $SL (2, ℂ)$ ⁠ are all of weight one.

Examples

Suppose $G = A \times B$ , and there are canonical irreducible representations $c A, c B$ of $A, B$ respectively. If $χ i, i = 1, \dots, k$ , are the irreducible representations of $A$ and $ψ j, j = 1, \dots, ℓ$ , are the irreducible representations of $B$ , then

χ_{i} \times ψ_{j} 1 \leq i \leq k, 1 \leq j \leq ℓ

are the irreducible representations of

A \times B

, where

χ_{i} \times ψ_{j} (a, b) = χ_{i} (a) ψ_{j} (b), (a, b) \in A \times B .

In this case, we have

⟨ (c_{A} \times c_{B}) \otimes (χ_{i} \times ψ_{ℓ}), χ_{n} \times ψ_{p} ⟩ = ⟨ c_{A} \otimes χ_{k}, χ_{n} ⟩ \cdot ⟨ c_{B} \otimes ψ_{ℓ}, ψ_{p} ⟩ .

Therefore, there is an arrow in the McKay graph of

G

between

χ_{i} \times ψ_{j}

and

χ_{k} \times ψ_{ℓ}

if and only if there is an arrow in the McKay graph of

A

between

χ i, χ k

and there is an arrow in the McKay graph of

B

between

ψ j, 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; all are conjugate to subgroups of ⁠ $SU (2, ℂ) .$ ⁠ 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, the binary tetrahedral group $\overline{T}$ is generated by the ⁠ $SU (2, ℂ)$ ⁠ matrices:

S = (\begin{matrix} i & 0 \\ 0 & - i \end{matrix}), V = (\begin{matrix} 0 & i \\ i & 0 \end{matrix}), U = \frac{1}{\sqrt{2}} (\begin{matrix} ε & ε^{3} \\ ε & ε^{7} \end{matrix}),

where

ε

is a primitive eighth root of unity. In fact, we have

\overline{T} = {U^{k}, S U^{k}, V U^{k}, S V U^{k} ∣ k = 0, \dots, 5} .

The conjugacy classes of

\overline{T}

are:

C_{1} = {U^{0} = I},

C_{2} = {U^{3} = - I},

C_{3} = {\pm S, \pm V, \pm 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

\overline{T}

is

Conjugacy Classes	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	$C_{6}$	$C_{7}$
$χ_{1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$χ_{2}$	$1$	$1$	$1$	$ω$	$ω^{2}$	$ω$	$ω^{2}$
$χ_{3}$	$1$	$1$	$1$	$ω^{2}$	$ω$	$ω^{2}$	$ω$
$χ_{4}$	$3$	$3$	$- 1$	$0$	$0$	$0$	$0$
$c$	$2$	$- 2$	$0$	$- 1$	$- 1$	$1$	$1$
$χ_{5}$	$2$	$- 2$	$0$	$- ω$	$- ω^{2}$	$ω$	$ω^{2}$
$χ_{6}$	$2$	$- 2$	$0$	$- ω^{2}$	$- ω$	$ω^{2}$	$ω$

Here

ω = e^{2 π i / 3} .

The canonical representation

V

is here denoted by

c

. Using the inner product, we find that the McKay graph of

\overline{T}

is the extended Coxeter–Dynkin diagram of type

{\tilde{E}}_{6} .

References

↑ Steinberg, Robert (1985), "Subgroups of $S U_{2}$ , Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics, 18: 587–598, doi:10.2140/pjm.1985.118.587
↑ McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag

McKay graph

Definition

Some results

Examples

See also

References

Further reading