McKay graph

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 of G, relative to V, as follows:

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

We can calculate the value of n 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 is defined to be the McKay graph of its canonical representation.

For finite subgroups of the canonical representation on is self-dual, so n_{ij}=n_{ji} for all i, j. Thus, the McKay graph of finite subgroups of is undirected.

In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of and the extended Coxeter-Dynkin diagrams of type A-D-E.

We define the Cartan matrix c of V as follows:

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

where δ 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 has no self-loops, that is, n_{ii}=0 for all i.
• The arrows of the McKay graph of a finite subgroup of are all of weight one.

Examples

• Suppose , and there are canonical irreducible representations cA, cB of A, B respectively. If , are the irreducible representations of A and , 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, χ and there is an arrow in the McKay graph of B between ψ**, ψ**. 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 are the binary polyhedral groups; all are conjugate to subgroups of 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 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.

References

References

1. Steinberg, Robert. (1985). "Subgroups of $SU\_2$, Dynkin diagrams and affine Coxeter elements". Pacific Journal of Mathematics.
2. McKay, John. (1982). ""The Geometric Vein", Coxeter Festschrift". [[Springer-Verlag]].