
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 formula_1 are irreducible representations of "G", then there is an arrow from formula_2 to formula_3 if and only if formula_3 is a constituent of the tensor product formula_5. Then the weight "nij" of the arrow is the number of times this constituent appears in formula_6. 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 formula_7, 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 formula_8 are the eigenvectors of "cV" to the eigenvalues formula_9, where formula_10 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 formula_11 be its character. Let formula_12 be the irreducible representations of "G". If
then define the McKay graph formula_14 of "G," relative to "V," as follows:
We can calculate the value of "nij" using inner product formula_21 on 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 on C2 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: 
where formula_24 is the Kronecker delta.