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

$V\otimes \chi _{i}=\sum _{j}n_{ij}\chi _{j},$

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

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
• 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, 37, Amer. Math. Soc.: 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