McKay graph

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 ${\displaystyle V\otimes \chi _{i}.}$ Then the weight n_ij of the arrow is the number of times this constituent appears in ${\displaystyle V\otimes \chi _{i}.}$ For finite subgroups H of ⁠${\displaystyle {\text{GL}}(2,\mathbb {C} ),}$⁠ the McKay graph of H is the McKay graph of the defining 2-dimensional representation of H.

Affine (extended) Dynkin diagrams

If G has n irreducible characters, then the Cartan matrix c_V 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 Robert 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 c_V to the eigenvalues ${\displaystyle d-\chi _{V}(g),}$ where χ_V is the character of the representation V.^[1]

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 ⁠${\displaystyle {\text{SL}}(2,\mathbb {C} )}$⁠ and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.^[2]

Definition

Let G be a finite group, V be a representation of G and χ be its character. Let ${\displaystyle \{\chi _{1},\ldots ,\chi _{d}\}}$ be the irreducible representations of G. If

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

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 ${\displaystyle \chi _{i}\xrightarrow {n_{ij}} \chi _{j},}$ or sometimes as n_ij unlabeled arrows.
• If ${\displaystyle n_{ij}=n_{ji},}$ we denote the two opposite arrows between χ _i, χ _j as an undirected edge of weight n_ij. Moreover, if ${\displaystyle n_{ij}=1,}$ we omit the weight label.

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

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

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

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

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

We define the Cartan matrix c_V of V as follows:

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

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

${\displaystyle \chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell }$

are the irreducible representations of A × 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 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 ⁠${\displaystyle {\text{SL}}(2,\mathbb {C} )}$⁠ are the binary polyhedral groups; all are conjugate to subgroups of ⁠${\displaystyle {\text{SU}}(2,\mathbb {C} ).}$⁠ 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 ${\displaystyle {\overline {T}}}$ is generated by the ⁠${\displaystyle {\text{SU}}(2,\mathbb {C} )}$⁠ matrices:

${\displaystyle S=\left({\begin{array}{cc}i&0\\0&-i\end{array}}\right),\ \ V=\left({\begin{array}{cc}0&i\\i&0\end{array}}\right),\ \ U={\frac {1}{\sqrt {2}}}\left({\begin{array}{cc}\varepsilon &\varepsilon ^{3}\\\varepsilon &\varepsilon ^{7}\end{array}}\right),}$

where ε is a primitive eighth root of unity. 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:

${\displaystyle C_{1}=\{U^{0}=I\},}$

${\displaystyle C_{2}=\{U^{3}=-I\},}$

${\displaystyle C_{3}=\{\pm S,\pm V,\pm SV\},}$

${\displaystyle C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},}$

${\displaystyle C_{5}=\{-U,SU,VU,SVU\},}$

${\displaystyle C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},}$

${\displaystyle C_{7}=\{U,-SU,-VU,-SVU\}.}$

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 V is here denoted by c. Using the inner product, we find that the McKay graph of ${\displaystyle {\overline {T}}}$ is the extended Coxeter–Dynkin diagram of type ${\displaystyle {\tilde {E}}_{6}.}$

See also

• ADE classification
• Binary tetrahedral group