# 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 $V\otimes \chi _{i}.$ Then the weight n$_{ij}$ of the arrow is the number of times this constituent appears in $V\otimes \chi _{i}.$ For finite subgroups H of ⁠${\text{GL}}(2,\mathbb {C} ),$⁠ 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 $c_{V}=(d\delta _{ij}-n_{ij})_{ij},$ 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 $((\chi _{i}(g))_{i}$ are the eigenvectors of c$_V$ to the eigenvalues $d-\chi _{V}(g),$ 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 ⁠${\text{SL}}(2,\mathbb {C} )$⁠ 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 $\{\chi _{1},\ldots ,\chi _{d}\}$ be the irreducible representations of G. If
    $V\otimes \chi _{i}=\sum \nolimits _{j}n_{ij}\chi _{j},$
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 $\chi _{i}\xrightarrow {n_{ij}} \chi _{j},$ or sometimes as n$_{ij}$ unlabeled arrows.
* If $n_{ij}=n_{ji},$ we denote the two opposite arrows between χ$_i$, χ$_j$ as an undirected edge of weight n$_{ij}$. Moreover, if $n_{ij}=1,$ we omit the weight label.
We can calculate the value of n$_{ij}$ using inner product $\langle \cdot ,\cdot \rangle$ on characters:
    $n_{ij}=\langle V\otimes \chi _{i},\chi _{j}\rangle ={\frac {1}{|G|}}\sum _{g\in G}V(g)\chi _{i}(g){\overline {\chi _{j}(g)}}.$
The McKay graph of a finite subgroup of ⁠${\text{GL}}(2,\mathbb {C} )$⁠ is defined to be the McKay graph of its canonical representation.
For finite subgroups of ⁠${\text{SL}}(2,\mathbb {C} ),$⁠ the canonical representation on ⁠$\mathbb {C} ^{2}$⁠ is self-dual, so $n_{ij}=n_{ji}$ for all i, j. Thus, the McKay graph of finite subgroups of ⁠${\text{SL}}(2,\mathbb {C} )$⁠ is undirected.
In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of ⁠${\text{SL}}(2,\mathbb {C} )$⁠ and the extended Coxeter-Dynkin diagrams of type A-D-E.
We define the Cartan matrix c$_V$ of V as follows:
    $c_{V}=(d\delta _{ij}-n_{ij})_{ij},$
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 ⁠${\text{SL}}(2,\mathbb {C} )$⁠ has no self-loops, that is, $n_{ii}=0$ for all i.
* The arrows of the McKay graph of a finite subgroup of ⁠${\text{SL}}(2,\mathbb {C} )$⁠ are all of weight one.

## Examples

* Suppose G = A × B, and there are canonical irreducible representations c$_A$, c$_B$ of A, 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
       $\chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell$
    are the irreducible representations of A × B, where $\chi _{i}\times \psi _{j}(a,b)=\chi _{i}(a)\psi _{j}(b),(a,b)\in A\times B.$ In this case, we have
       $\langle (c_{A}\times c_{B})\otimes (\chi _{i}\times \psi _{\ell }),\chi _{n}\times \psi _{p}\rangle =\langle c_{A}\otimes \chi _{k},\chi _{n}\rangle \cdot \langle c_{B}\otimes \psi _{\ell },\psi _{p}\rangle .$
    Therefore, there is an arrow in the McKay graph of G between $\chi _{i}\times \psi _{j}$ and $\chi _{k}\times \psi _{\ell }$ 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 ⁠${\text{SL}}(2,\mathbb {C} )$⁠ are the binary polyhedral groups; all are conjugate to subgroups of ⁠${\text{SU}}(2,\mathbb {C} ).$⁠ 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 ⁠${\text{SU}}(2,\mathbb {C} )$⁠ matrices:
       $S=\left({\begin{array}{cc}i&0\\0&-i\end{array}}\right),\ \ V=\left({\begin{array}{cc}0&i\\i&0\end{array}}\right),\ \ U={\frac {1}{\sqrt {2}}}\left({\begin{array}{cc}\varepsilon &\varepsilon ^{3}\\\varepsilon &\varepsilon ^{7}\end{array}}\right),$
    where ε is a primitive eighth root of unity. In fact, we have
       ${\overline {T}}=\{U^{k},SU^{k},VU^{k},SVU^{k}\mid k=0,\ldots ,5\}.$
    The conjugacy classes of ${\overline {T}}$ are:
       $C_{1}=\{U^{0}=I\},$
       $C_{2}=\{U^{3}=-I\},$
       $C_{3}=\{\pm S,\pm V,\pm SV\},$
       $C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},$
       $C_{5}=\{-U,SU,VU,SVU\},$
       $C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},$
       $C_{7}=\{U,-SU,-VU,-SVU\}.$
    The character table of ${\overline {T}}$ is
| Conjugacy Classes | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$ | $C_{7}$ |
| $\chi _{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi _{2}$ | $1$ | $1$ | $1$ | $\omega$ | $\omega ^{2}$ | $\omega$ | $\omega ^{2}$ |
| $\chi _{3}$ | $1$ | $1$ | $1$ | $\omega ^{2}$ | $\omega$ | $\omega ^{2}$ | $\omega$ |
| $\chi _{4}$ | $3$ | $3$ | $-1$ | $0$ | $0$ | $0$ | $0$ |
| $c$ | $2$ | $-2$ | $0$ | $-1$ | $-1$ | $1$ | $1$ |
| $\chi _{5}$ | $2$ | $-2$ | $0$ | $-\omega$ | $-\omega ^{2}$ | $\omega$ | $\omega ^{2}$ |
| $\chi _{6}$ | $2$ | $-2$ | $0$ | $-\omega ^{2}$ | $-\omega$ | $\omega ^{2}$ | $\omega$ |
    Here $\omega =e^{2\pi i/3}.$ 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 ${\tilde {E}}_{6}.$