McKay graph

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 $χi, χj$ are irreducible representations of $G$, then there is an arrow from $χi$ to $χj$ if and only if $χj$ is a constituent of the tensor product $$V\otimes\chi_i.$$ Then the weight $nij$ of the arrow is the number of times this constituent appears in $$V \otimes\chi_i.$$ For finite subgroups $H$ of $\text{GL}(2, \C),$ 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 $cV$ 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 $cV$ to the eigenvalues $$ d-\chi_V(g),$$ where $χV$ 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 $\text{SL}(2, \C)$ 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


 * $$V\otimes\chi_i = \sum_j n_{ij} \chi_j,$$

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


 * Each irreducible representation of $G$ corresponds to a node in $ΓG$.
 * If $nij > 0$, there is an arrow from $χi$ to $χj$ of weight $nij$, written as $$\chi_i\xrightarrow{n_{ij}}\chi_j,$$ or sometimes as $nij$ unlabeled arrows.
 * If $$n_{ij} = n_{ji},$$ we denote the two opposite arrows between $χi, χj$ as an undirected edge of weight $nij$. Moreover, if $$n_{ij} = 1,$$ we omit the weight label.

We can calculate the value of $nij$ 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 $\text{GL}(2, \C)$ is defined to be the McKay graph of its canonical representation.

For finite subgroups of $\text{SL}(2, \C),$ the canonical representation on $\C^2$ is self-dual, so $$n_{ij}=n_{ji}$$ for all $i, j$. Thus, the McKay graph of finite subgroups of $\text{SL}(2, \C)$ is undirected.

In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of $\text{SL}(2, \C)$ and the extended Coxeter-Dynkin diagrams of type A-D-E.

We define the Cartan matrix $cV$ of $V$ as follows:


 * $$c_V = (d\delta_{ij} - n_{ij})_{ij},$$

where $δij$ 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 $\text{SL}(2, \C)$ has no self-loops, that is, $$n_{ii}=0$$ for all $i$.
 * The arrows of the McKay graph of a finite subgroup of $\text{SL}(2, \C)$ 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 $χi, i = 1, …, k$, are the irreducible representations of $A$ and $ψj, j = 1, …, ℓ$, 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 $χi, χk$ and there is an arrow in the McKay graph of $B$ between $ψj, ψℓ$. 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 $\text{SL}(2, \C)$ are the binary polyhedral groups; all are conjugate to subgroups of $\text{SU}(2, \C).$ 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 $\text{SU}(2, \C)$ 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


 * 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 $$\tilde{E}_6.$$