McKay graph explained

V ⊗ \chi _{i.} Then the weight of the arrow is the number of times this constituent appears in V ⊗ \chi _{i.} For finite subgroups of the McKay graph of is the McKay graph of the defining 2-dimensional representation of .

If has irreducible characters, then the Cartan matrix of the representation of dimension is defined by

c _{V} = (d\delta _{ij} - n _{ij} ) _{ij} , where is the Kronecker delta. A result by Robert Steinberg states that if is a representative of a conjugacy class of, then the vectors ((\chi _{i(g))} _{i} are the eigenvectors of to the eigenvalues d - \chi _{V(g),} where is the character of the representation .

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 and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.

Definition

Let be a finite group, be a representation of and be its character. Let

\{\chi _{1,\ldots,\chi} _{d\}} be the irreducible representations of . IfV ⊗ \chi _{i} = \sum _{j} n _{ij} \chi _{j,}

then define the McKay graph of, relative to, as follows:

• Each irreducible representation of corresponds to a node in .
• If, there is an arrow from to of weight, written as

\chi _{i\xrightarrow{n} _{ij} }\chi_j, or sometimes as unlabeled arrows.

• If

n _{ij} = n _{ji} , we denote the two opposite arrows between as an undirected edge of weight . Moreover, if n _{ij} = 1 , we omit the weight label.\langle ⋅ , ⋅ \rangle on characters: n _{ij} = \langle V ⊗ \chi _{i,} \chi _{j\rangle} = (1)/(|G|) \sum _{g\in} V(g)\chi _{i(g)\overline{\chi} _{j(g)}.}

The McKay graph of a finite subgroup of is defined to be the McKay graph of its canonical representation.

For finite subgroups of the canonical representation on is self-dual, so

n _{ij} = n _{ji} for all . Thus, the McKay graph of finite subgroups of is undirected.

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

We define the Cartan matrix of as follows:

c _{V} = (d\delta _{ij} - n _{ij} ) _{ij} ,

where is the Kronecker delta.

Some results

• If the representation is faithful, then every irreducible representation is contained in some tensor power

V ^{⊗} , and the McKay graph of is connected.

• The McKay graph of a finite subgroup of has no self-loops, that is,

n _{ii} =0 for all .

• The arrows of the McKay graph of a finite subgroup of are all of weight one.

Examples

• Suppose, and there are canonical irreducible representations of respectively. If, are the irreducible representations of and, are the irreducible representations of, then

\chi _{i x \psi} _{j} 1 \leq i \leq k, 1 \leq j \leq \ell

are the irreducible representations of, where

\chi _{i x \psi} _{j(a,b)} = \chi _{i(a)\psi} _{j(b),} (a,b)\in A x B. In this case, we have \langle (c _{A x} c _{B) ⊗} (\chi _{i x \psi} _{\ell),} \chi _{n x \psi} _{p\rangle} = \langle c _{A ⊗} \chi _{k,} \chi _{n\rangle ⋅} \langle c _{B ⊗} \psi _{\ell,} \psi _{p\rangle.}

Therefore, there is an arrow in the McKay graph of between

\chi _{i x \psi} _{j} and \chi _{k x \psi} _{\ell} if and only if there is an arrow in the McKay graph of between and there is an arrow in the McKay graph of between . In this case, the weight on the arrow in the McKay graph of is the product of the weights of the two corresponding arrows in the McKay graphs of and .\overline{T} is generated by the 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 = (1)/(\sqrt{ 2) } \left(\begin\varepsilon & \varepsilon^3 \\\varepsilon & \varepsilon^7 \end \right),

where 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

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

See also

• ADE classification
• Binary tetrahedral group