# McKay graph

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 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 character of G. If $\chi_i, \chi_j$ are irreducible representations of G then there is an arrow from $\chi_i$ to $\chi_j$ if and only if $\chi_j$ 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 GL(2, C), the McKay graph of H is the McKay graph of the canonical 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 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 $\chi_V$ 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 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 $\chi$ 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 $\Gamma_G$ of G as follow:

• To each irreducible representation of G corresponds a node in $\Gamma_G$.
• There is an arrow from $\chi_i$ to $\chi_j$ if and only if nij > 0 and nij is the weight of the arrow: $\chi_i\xrightarrow{n_{ij}}\chi_j$.
• If nij = nji, then we put an edge between $\chi_i$ and $\chi_j$ instead of a double arrow. Moreover, if nij = 1, then we do not write the weight of the corresponding arrow.

We can calculate the value of nij by considering the inner product. We have the following formula:

$n_{ij} = \langle V\otimes\chi_i, \chi_j\rangle = \frac{1}{|G|}\sum_{g\in G} V(g)\chi_i(g)\overline{\chi_j(g)},$

where $\langle \cdot, \cdot \rangle$ denotes the inner product of the characters.

The McKay graph of a finite subgroup of GL(2, C) is defined to be the McKay graph of its canonical representation.

For finite subgroups of SL(2, C), the canonical representation is self-dual, so nij = nji for all i, j. Thus, the McKay graph of finite subgroups of SL(2, C) is undirected.

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

We define the Cartan matrix cV of V as follow:

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

where $\delta_{ij}$ is the Kronecker delta.

## Some results

• If the representation V of a finite group G is faithful, then the McKay graph of V is connected.
• The McKay graph of a finite subgroup of SL(2, C) has no self-loops, that is, nii = 0 for all i.
• The weights of the arrows of the McKay graph of a finite subgroup of SL(2, C) are always less or equal than one.

## Examples

• Suppose G = A × B, and there are canonical irreducible representations cA and cB of A and B respectively. If $\chi_i$, i = 1, ..., k, are the irreducible representations of A and $\psi_j$, j = 1, ..., l, are the irreducible representations of B, then
$\chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq l$

are the irreducible representations of $A\times B$, 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

$\langle (c_A\times c_B)\otimes (\chi_i\times\psi_l), \chi_n\times\psi_p\rangle = \langle c_A\otimes \chi_k, \chi_n\rangle\cdot \langle c_B\otimes \psi_l, \psi_p\rangle.$

Therefore, there is an arrow in the McKay graph of G between $\chi_i\times\psi_j$ and $\chi_k\times\psi_l$ if and only if there is an arrow in the McKay graph of A between $\chi_i$ and $\chi_k$ and there is an arrow in the McKay graph of B between $\psi_j$ and $\psi_l$. 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 SL(2, C) are the binary polyhedral groups. 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, let $\overline{T}$ be the binary tetrahedral group. Every finite subgroup of SL(2, C) is conjugate to a finite subgroup of SU(2, C). Consider the matrices in SU(2, C):
$S = \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right) , V = \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right), U = \frac{1}{\sqrt{2}} \left( \begin{array}{cc} \epsilon & \epsilon^3 \\ \epsilon & \epsilon^7 \end{array} \right),$

where ε is a primitive eighth root of unity. Then, $\overline{T}$ is generated by S, U, V. In fact, we have

$\overline{T} = \{U^k, SU^k,VU^k,SVU^k | k = 0,\ldots, 5\}.$

The conjugacy classes of $\overline{T}$ are the following:

$C_1 = \{U^0 = I\},$
$C_2 = \{U^3 = - I\},$
$C_3 = \{\pm S, \pm V, \pm SV\},$
$C_4 = \{U^2, SU^2, VU^2, SVU^2\},$
$C_5 = \{-U, SU, VU, SVU\},$
$C_6 = \{-U^2, -SU^2, -VU^2, -SVU^2\},$
$C_7 = \{U, -SU, -VU, -SVU\}.$

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 is represented by c. By using the inner product, we have that the McKay graph of $\overline{T}$ is the extended Coxeter-Dynkin diagram of type $\tilde{E}_6$.

## References

• 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 fünten Grade", Teubner (Leibniz)
• McKay, John (1980), "Graphs, singularities and finite groups", Proc. Symp. Pure Math. (Amer. Math. Soc.) 37: 183–186, doi:10.1090/pspum/037/604577
• 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 $SU_2$, Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics 18: 587–598