McKay graph

Short description: Construction in graph theory

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 [math]\displaystyle{ V\otimes\chi_i. }[/math] Then the weight n_ij of the arrow is the number of times this constituent appears in [math]\displaystyle{ V \otimes\chi_i. }[/math] For finite subgroups H of [math]\displaystyle{ \text{GL}(2, \C), }[/math] 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 [math]\displaystyle{ c_V = (d\delta_{ij} -n_{ij})_{ij} , }[/math] where δ is the Kronecker delta. A result by (Steinberg 1985) states that if g is a representative of a conjugacy class of G, then the vectors [math]\displaystyle{ ((\chi_i(g))_i }[/math] are the eigenvectors of c_V to the eigenvalues [math]\displaystyle{ d-\chi_V(g), }[/math] where χ_V is the character of the representation V.

The McKay correspondence (McKay 1982), named after John McKay, states that there is a one-to-one correspondence between the McKay graphs of the finite subgroups of [math]\displaystyle{ \text{SL}(2, \C) }[/math] 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 [math]\displaystyle{ \{\chi_1,\ldots,\chi_d\} }[/math] be the irreducible representations of G. If

[math]\displaystyle{ V\otimes\chi_i = \sum_j n_{ij} \chi_j, }[/math]

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 [math]\displaystyle{ \chi_i\xrightarrow{n_{ij}}\chi_j, }[/math] or sometimes as n_ij unlabeled arrows.
• If [math]\displaystyle{ n_{ij} = n_{ji}, }[/math] we denote the two opposite arrows between χ _i, χ _j as an undirected edge of weight n_ij. Moreover, if [math]\displaystyle{ n_{ij} = 1, }[/math] we omit the weight label.

We can calculate the value of n_ij using inner product [math]\displaystyle{ \langle \cdot, \cdot \rangle }[/math] on characters:

[math]\displaystyle{ 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)}. }[/math]

The McKay graph of a finite subgroup of [math]\displaystyle{ \text{GL}(2, \C) }[/math] is defined to be the McKay graph of its canonical representation.

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

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

We define the Cartan matrix c_V of V as follows:

[math]\displaystyle{ c_V = (d\delta_{ij} - n_{ij})_{ij}, }[/math]

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 [math]\displaystyle{ V^{\otimes k}, }[/math] and the McKay graph of V is connected.
• The McKay graph of a finite subgroup of [math]\displaystyle{ \text{SL}(2, \C) }[/math] has no self-loops, that is, [math]\displaystyle{ n_{ii}=0 }[/math] for all i.
• The arrows of the McKay graph of a finite subgroup of [math]\displaystyle{ \text{SL}(2, \C) }[/math] 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

[math]\displaystyle{ \chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq \ell }[/math]

are the irreducible representations of A × B, where [math]\displaystyle{ \chi_i\times\psi_j(a,b) = \chi_i(a)\psi_j(b), (a,b)\in A\times B. }[/math] In this case, we have

[math]\displaystyle{ \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. }[/math]

Therefore, there is an arrow in the McKay graph of G between [math]\displaystyle{ \chi_i\times\psi_j }[/math] and [math]\displaystyle{ \chi_k\times\psi_\ell }[/math] 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 [math]\displaystyle{ \text{SL}(2, \C) }[/math] are the binary polyhedral groups; all are conjugate to subgroups of [math]\displaystyle{ \text{SU}(2, \C). }[/math] 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 [math]\displaystyle{ \overline{T} }[/math] is generated by the [math]\displaystyle{ \text{SU}(2, \C) }[/math] matrices:

[math]\displaystyle{ 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), }[/math]

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

[math]\displaystyle{ \overline{T} = \{U^k, SU^k,VU^k,SVU^k \mid k = 0,\ldots, 5\}. }[/math]

The conjugacy classes of [math]\displaystyle{ \overline{T} }[/math] are:

[math]\displaystyle{ C_1 = \{U^0 = I\}, }[/math]

[math]\displaystyle{ C_2 = \{U^3 = - I\}, }[/math]

[math]\displaystyle{ C_3 = \{\pm S, \pm V, \pm SV\}, }[/math]

[math]\displaystyle{ C_4 = \{U^2, SU^2, VU^2, SVU^2\}, }[/math]

[math]\displaystyle{ C_5 = \{-U, SU, VU, SVU\}, }[/math]

[math]\displaystyle{ C_6 = \{-U^2, -SU^2, -VU^2, -SVU^2\}, }[/math]

[math]\displaystyle{ C_7 = \{U, -SU, -VU, -SVU\}. }[/math]

The character table of [math]\displaystyle{ \overline{T} }[/math] is

Conjugacy Classes	[math]\displaystyle{ C_1 }[/math]	[math]\displaystyle{ C_2 }[/math]	[math]\displaystyle{ C_3 }[/math]	[math]\displaystyle{ C_4 }[/math]	[math]\displaystyle{ C_5 }[/math]	[math]\displaystyle{ C_6 }[/math]	[math]\displaystyle{ C_7 }[/math]
[math]\displaystyle{ \chi_1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]
[math]\displaystyle{ \chi_2 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ \omega }[/math]	[math]\displaystyle{ \omega^2 }[/math]	[math]\displaystyle{ \omega }[/math]	[math]\displaystyle{ \omega^2 }[/math]
[math]\displaystyle{ \chi_3 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ \omega^2 }[/math]	[math]\displaystyle{ \omega }[/math]	[math]\displaystyle{ \omega^2 }[/math]	[math]\displaystyle{ \omega }[/math]
[math]\displaystyle{ \chi_4 }[/math]	[math]\displaystyle{ 3 }[/math]	[math]\displaystyle{ 3 }[/math]	[math]\displaystyle{ -1 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ 0 }[/math]
[math]\displaystyle{ c }[/math]	[math]\displaystyle{ 2 }[/math]	[math]\displaystyle{ -2 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ -1 }[/math]	[math]\displaystyle{ -1 }[/math]	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 1 }[/math]
[math]\displaystyle{ \chi_5 }[/math]	[math]\displaystyle{ 2 }[/math]	[math]\displaystyle{ -2 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ -\omega }[/math]	[math]\displaystyle{ -\omega^2 }[/math]	[math]\displaystyle{ \omega }[/math]	[math]\displaystyle{ \omega^2 }[/math]
[math]\displaystyle{ \chi_6 }[/math]	[math]\displaystyle{ 2 }[/math]	[math]\displaystyle{ -2 }[/math]	[math]\displaystyle{ 0 }[/math]	[math]\displaystyle{ -\omega^2 }[/math]	[math]\displaystyle{ -\omega }[/math]	[math]\displaystyle{ \omega^2 }[/math]	[math]\displaystyle{ \omega }[/math]

Here [math]\displaystyle{ \omega = e^{2\pi i/3}. }[/math] The canonical representation V is here denoted by c. Using the inner product, we find that the McKay graph of [math]\displaystyle{ \overline{T} }[/math] is the extended Coxeter–Dynkin diagram of type [math]\displaystyle{ \tilde{E}_6. }[/math]

See also

• ADE classification
• Binary tetrahedral group

References

• 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 
• McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag 
• 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 
• Steinberg, Robert (1985), "Subgroups of [math]\displaystyle{ SU_2 }[/math], Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics 18: 587–598, doi:10.2140/pjm.1985.118.587