McKay graph

Short description: Construction in graph theory

320px 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 $\{{\chi }_1,\dots ,{\chi }_d\}$ be the irreducible representations of G. If

$V\otimes {\chi }_i=\sum _j{n}_{ij}{\chi }_j,$

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 ${\chi }_i\stackrel{n_{ij}}{\to }{\chi }_j,$ 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 $⟨\cdot ,\cdot ⟩$ on characters:

$n_{ij}=⟨V\otimes {\chi }_i,{\chi }_j⟩=\frac{1}{|G|}\sum _{g\in G}V(g){\chi }_i(g)\overline{{\chi }_j(g)}.$

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

${\chi }_i\times {\psi }_{j}1\le i\le k,1\le j\le \ell$

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

$⟨(c_A\times c_B)\otimes ({\chi }_i\times {\psi }_{\ell }),{\chi }_n\times {\psi }_p⟩=⟨c_A\otimes {\chi }_k,{\chi }_n⟩\cdot ⟨c_B\otimes {\psi }_{\ell },{\psi }_p⟩.$

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 $\bar{T}$ is generated by the $\text{SU}(2,\mathbb{ℂ})$ matrices:

$S=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),V=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),U=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\epsilon & {\epsilon }^3\\ \epsilon & {\epsilon }^7\end{array}\right),$

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

$\bar{T}=\{U^k,S{U}^k,V{U}^k,SV{U}^k\mid k=0,\dots ,5\}.$

The conjugacy classes of $\bar{T}$ are:

$C_1=\{U^0=I\},$

$C_2=\{U^3=-I\},$

$C_3=\{\pm S,\pm V,\pm SV\},$

$C_4=\{U^2,S{U}^2,V{U}^2,SV{U}^2\},$

$C_5=\{-U,SU,VU,SVU\},$

$C_6=\{-U^2,-S{U}^2,-V{U}^2,-SV{U}^2\},$

$C_7=\{U,-SU,-VU,-SVU\}.$

The character table of $\bar{T}$ 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 $\bar{T}$ is the extended Coxeter–Dynkin diagram of type ${\tilde{E}}_6.$

See also

• ADE classification
• Binary tetrahedral group

Further reading

• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7, https://archive.org/details/introductiontoli00jame 
• James, Gordon; Liebeck, Martin (2001), Representations and Characters of Groups (2nd ed.), Cambridge University Press, ISBN 0-521-00392-X 
• Klein, Felix (1884), "Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade", Teubner (Leibniz) 
• McKay, John (1980), "Graphs, singularities and finite groups", Proc. Symp. Pure Math., Proceedings of Symposia in Pure Mathematics (Amer. Math. Soc.) 37: 183–186, doi:10.1090/pspum/037/604577, ISBN 9780821814406 
• Riemenschneider, Oswald (2005), McKay correspondence for quotient surface singularities, Singularities in Geometry and Topology, Proceedings of the Trieste Singularity Summer School and Workshop, pp. 483–519