McKay graph

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 ${\displaystyle V\otimes \chi _{i}.}$ 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.

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

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

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

References

Further reading

• 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., Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc., 37: 183–186, doi:10.1090/pspum/037/604577, ISBN 9780821814406
• 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

This page was last edited on 21 March 2024, at 06:33