# McKay graph

(Redirected from McKay correspondence)
 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 ${\displaystyle \chi _{i},\chi _{j}}$ are irreducible representations of G then there is an arrow from ${\displaystyle \chi _{i}}$ to ${\displaystyle \chi _{j}}$ if and only if ${\displaystyle \chi _{j}}$ is a constituent of the tensor product ${\displaystyle V\otimes \chi _{i}}$. Then the weight nij of the arrow is the number of times this constituent appears in ${\displaystyle V\otimes \chi _{i}}$. 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 ${\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij}}$, 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 ${\displaystyle ((\chi _{i}(g))_{i}}$ are the eigenvectors of cV to the eigenvalues ${\displaystyle d-\chi _{V}(g)}$, where ${\displaystyle \chi _{V}}$ 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 ${\displaystyle \chi }$ be its character. Let ${\displaystyle \{\chi _{1},\ldots ,\chi _{d}\}}$ be the irreducible representations of G. If

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

then define the McKay graph ${\displaystyle \Gamma _{G}}$ of G as follows:

• To each irreducible representation of G corresponds a node in ${\displaystyle \Gamma _{G}}$.
• There is an arrow from ${\displaystyle \chi _{i}}$ to ${\displaystyle \chi _{j}}$ if and only if nij > 0 and nij is the weight of the arrow: ${\displaystyle \chi _{i}{\xrightarrow {n_{ij}}}\chi _{j}}$.
• If nij = nji, then we put an edge between ${\displaystyle \chi _{i}}$ and ${\displaystyle \chi _{j}}$ 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:

${\displaystyle n_{ij}=\langle V\otimes \chi _{i},\chi _{j}\rangle ={\frac {1}{|G|}}\sum _{g\in G}V(g)\chi _{i}(g){\overline {\chi _{j}(g)}},}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 follows:

${\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},}$

where ${\displaystyle \delta _{ij}}$ 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 ${\displaystyle \chi _{i}}$, i = 1, ..., k, are the irreducible representations of A and ${\displaystyle \psi _{j}}$, j = 1, ..., , are the irreducible representations of B, then
${\displaystyle \chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell }$
are the irreducible representations of ${\displaystyle A\times B}$, where ${\displaystyle \chi _{i}\times \psi _{j}(a,b)=\chi _{i}(a)\psi _{j}(b),(a,b)\in A\times B}$. In this case, we have
${\displaystyle \langle (c_{A}\times c_{B})\otimes (\chi _{i}\times \psi _{\ell }),\chi _{n}\times \psi _{p}\rangle =\langle c_{A}\otimes \chi _{k},\chi _{n}\rangle \cdot \langle c_{B}\otimes \psi _{\ell },\psi _{p}\rangle .}$
Therefore, there is an arrow in the McKay graph of G between ${\displaystyle \chi _{i}\times \psi _{j}}$ and ${\displaystyle \chi _{k}\times \psi _{\ell }}$ if and only if there is an arrow in the McKay graph of A between ${\displaystyle \chi _{i}}$ and ${\displaystyle \chi _{k}}$ and there is an arrow in the McKay graph of B between ${\displaystyle \psi _{j}}$ and ${\displaystyle \psi _{\ell }}$. 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 ${\displaystyle {\overline {T}}}$ 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):
${\displaystyle S=\left({\begin{array}{cc}i&0\\0&-i\end{array}}\right),V=\left({\begin{array}{cc}0&i\\i&0\end{array}}\right),U={\frac {1}{\sqrt {2}}}\left({\begin{array}{cc}\varepsilon &\varepsilon ^{3}\\\varepsilon &\varepsilon ^{7}\end{array}}\right),}$
where ε is a primitive eighth root of unity. Then, ${\displaystyle {\overline {T}}}$ is generated by S, U, V. In fact, we have
${\displaystyle {\overline {T}}=\{U^{k},SU^{k},VU^{k},SVU^{k}\mid k=0,\ldots ,5\}.}$
The conjugacy classes of ${\displaystyle {\overline {T}}}$ are the following:
${\displaystyle C_{1}=\{U^{0}=I\},}$
${\displaystyle C_{2}=\{U^{3}=-I\},}$
${\displaystyle C_{3}=\{\pm S,\pm V,\pm SV\},}$
${\displaystyle C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},}$
${\displaystyle C_{5}=\{-U,SU,VU,SVU\},}$
${\displaystyle C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},}$
${\displaystyle C_{7}=\{U,-SU,-VU,-SVU\}.}$
The character table of ${\displaystyle {\overline {T}}}$ is
Conjugacy Classes ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle C_{3}}$ ${\displaystyle C_{4}}$ ${\displaystyle C_{5}}$ ${\displaystyle C_{6}}$ ${\displaystyle C_{7}}$
${\displaystyle \chi _{1}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle \chi _{2}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle \omega }$ ${\displaystyle \omega ^{2}}$ ${\displaystyle \omega }$ ${\displaystyle \omega ^{2}}$
${\displaystyle \chi _{3}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle \omega ^{2}}$ ${\displaystyle \omega }$ ${\displaystyle \omega ^{2}}$ ${\displaystyle \omega }$
${\displaystyle \chi _{4}}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle -1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$
${\displaystyle c}$ ${\displaystyle 2}$ ${\displaystyle -2}$ ${\displaystyle 0}$ ${\displaystyle -1}$ ${\displaystyle -1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle \chi _{5}}$ ${\displaystyle 2}$ ${\displaystyle -2}$ ${\displaystyle 0}$ ${\displaystyle -\omega }$ ${\displaystyle -\omega ^{2}}$ ${\displaystyle \omega }$ ${\displaystyle \omega ^{2}}$
${\displaystyle \chi _{6}}$ ${\displaystyle 2}$ ${\displaystyle -2}$ ${\displaystyle 0}$ ${\displaystyle -\omega ^{2}}$ ${\displaystyle -\omega }$ ${\displaystyle \omega ^{2}}$ ${\displaystyle \omega }$
Here ${\displaystyle \omega =e^{2\pi i/3}}$. The canonical representation is represented by c. By using the inner product, we have that the McKay graph of ${\displaystyle {\overline {T}}}$ is the extended Coxeter–Dynkin diagram of type ${\displaystyle {\tilde {E}}_{6}}$.

## 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 fünften 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 ${\displaystyle SU_{2}}$, Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics, 18: 587–598