McKay graph

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Affine (extended) Dynkin diagrams

In mathematics, the McKay graph of a finite-dimensional representation {{mvar|V}} of a finite group {{mvar|G}} is a weighted quiver encoding the structure of the representation theory of {{mvar|G}}. Each node represents an irreducible representation of {{mvar|G}}. If {{math|χ{{sub| i}}, χ{{sub| j}}}} are irreducible representations of {{mvar|G}}, then there is an arrow from {{math|χ{{sub| i}}}} to {{math|χ{{sub| j}}}} if and only if {{math|χ{{sub| j}}}} is a constituent of the tensor product $V\otimes\chi_i.$ Then the weight {{mvar|n{{sub|ij}}}} of the arrow is the number of times this constituent appears in $V \otimes\chi_i.$ For finite subgroups {{mvar|H}} of {{tmath|\text{GL}(2, \C),}} the McKay graph of {{mvar|H}} is the McKay graph of the defining 2-dimensional representation of {{mvar|H}}.

If {{mvar|G}} has {{mvar|n}} irreducible characters, then the Cartan matrix {{mvar|c{{sub|V}}}} of the representation {{mvar|V}} of dimension {{mvar|d}} is defined by $c_V = (d\delta_{ij} -n_{ij})_{ij} ,$ where {{math|δ}} is the Kronecker delta. A result by Robert Steinberg states that if {{mvar|g}} is a representative of a conjugacy class of {{mvar|G}}, then the vectors $((\chi_i(g))_i$ are the eigenvectors of {{mvar|c{{sub|V}}}} to the eigenvalues $d-\chi_V(g),$ where {{mvar|χ{{sub|V}}}} is the character of the representation {{mvar|V}}.{{Citation | first = Robert | last = Steinberg | title = Subgroups of $SU_2$ , Dynkin diagrams and affine Coxeter elements| year = 1985 |journal = Pacific Journal of Mathematics| volume = 18 | pages = 587–598| doi = 10.2140/pjm.1985.118.587 }}

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 {{tmath|\text{SL}(2, \C)}} and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.{{Citation | first = John | last = McKay | authorlink = John McKay (mathematician) | chapter = Representations and Coxeter Graphs |title = "The Geometric Vein", Coxeter Festschrift | year = 1982 | publisher = Springer-Verlag | location = Berlin}}

Definition

Let {{mvar|G}} be a finite group, {{mvar|V}} be a representation of {{mvar|G}} and {{mvar|χ}} be its character. Let $\{\chi_1,\ldots,\chi_d\}$ be the irreducible representations of {{mvar|G}}. If

: $V\otimes\chi_i = \sum\nolimits_j n_{ij} \chi_j,$

then define the McKay graph {{math|Γ{{sub|G}}}} of {{mvar|G}}, relative to {{mvar|V}}, as follows:

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

We can calculate the value of {{mvar|n{{sub|ij}}}} using inner product $\langle \cdot, \cdot \rangle$ on characters:

: $n_{ij} = \langle V\otimes\chi_i, \chi_j\rangle = \frac{1}$

\sum_{g\in G} V(g)\chi_i(g)\overline{\chi_j(g)}.

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

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

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

We define the Cartan matrix {{mvar|c{{sub|V}}}} of {{mvar|V}} as follows:

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

where {{mvar|δ{{sub|ij}}}} is the Kronecker delta.

Some results

If the representation {{mvar|V}} is faithful, then every irreducible representation is contained in some tensor power $V^{\otimes k},$ and the McKay graph of {{mvar|V}} is connected.
The McKay graph of a finite subgroup of {{tmath|\text{SL}(2, \C)}} has no self-loops, that is, $n_{ii}=0$ for all {{mvar|i}}.
The arrows of the McKay graph of a finite subgroup of {{tmath|\text{SL}(2, \C)}} are all of weight one.

Examples

Suppose {{math|1=G = A × B}}, and there are canonical irreducible representations {{mvar|c_A, c_B}} of {{mvar|A, B}} respectively. If {{math|1=χ{{sub| i}}, i = 1, …, k}}, are the irreducible representations of {{mvar|A}} and {{math|1=ψ{{sub| j}}, j = 1, …, ℓ}}, are the irreducible representations of {{mvar|B}}, then

:: $\chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq \ell$

: are the irreducible representations of {{math|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 {{mvar|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 {{mvar|A}} between {{mvar|χ{{sub|i}}, χ{{sub|k}}}} and there is an arrow in the McKay graph of {{mvar|B}} between {{math|ψ{{sub| j}}, ψ{{sub|ℓ}}}}. In this case, the weight on the arrow in the McKay graph of {{mvar|G}} is the product of the weights of the two corresponding arrows in the McKay graphs of {{mvar|A}} and {{mvar|B}}.

Felix Klein proved that the finite subgroups of {{tmath|\text{SL}(2, \C)}} are the binary polyhedral groups; all are conjugate to subgroups of {{tmath|\text{SU}(2, \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 $\overline{T}$ is generated by the {{tmath|\text{SU}(2, \C)}} 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 {{mvar|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

class="wikitable" border="1" ! Conjugacy Classes !! $C_1$ !! $C_2$ !! $C_3$ !! $C_4$ !! $C_5$ !! $C_6$ !! $C_7$
$\chi_1$ \| $1$ \| $1$ \| $1$ \| $1$ \| $1$ \| $1$ \| $1$
$\chi_2$ \| $1$ \| $1$ \| $1$ \| $\omega$ \| $\omega^2$ \| $\omega$ \| $\omega^2$
$\chi_3$ \| $1$ \| $1$ \| $1$ \| $\omega^2$ \| $\omega$ \| $\omega^2$ \| $\omega$
$\chi_4$ \| $3$ \| $3$ \| $-1$ \| $0$ \| $0$ \| $0$ \| $0$
$c$ \| $2$ \| $-2$ \| $0$ \| $-1$ \| $-1$ \| $1$ \| $1$
$\chi_5$ \| $2$ \| $-2$ \| $0$ \| $-\omega$ \| $-\omega^2$ \| $\omega$ \| $\omega^2$
$\chi_6$ \| $2$ \| $-2$ \| $0$ \| $-\omega^2$ \| $-\omega$ \| $\omega^2$ \| $\omega$

class="wikitable" border="1"

! Conjugacy Classes !! $C_1$ !! $C_2$ !! $C_3$ !! $C_4$ !! $C_5$ !! $C_6$ !! $C_7$

\chi_1

| $1$

\chi_2

| $1$

| $\omega$

| $\omega^2$

| $\omega$

| $\omega^2$

\chi_3

| $1$

| $\omega^2$

| $\omega$

| $\omega^2$

| $\omega$

\chi_4

| $3$

| $-1$

| $0$

c

| $2$

| $-2$

| $0$

| $-1$

| $1$

\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 {{mvar|V}} is here denoted by {{mvar|c}}. Using the inner product, we find that the McKay graph of $\overline{T}$ is the extended Coxeter–Dynkin diagram of type $\tilde{E}_6.$

McKay graph

Definition

Some results

Examples

See also

References

Further reading