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 are irreducible representations of G, then there is an arrow from to if and only if is a constituent of the tensor product. Then the weight nij of the arrow is the number of times this constituent appears in. For finite subgroups H of GL, 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, 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 are the eigenvectors of cV to the eigenvalues, where 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 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 be its character. Let be the irreducible representations of G. If
then define the McKay graph of G, relative to V, as follows:
We can calculate the value of nij using inner product on characters:
The McKay graph of a finite subgroup of GL is defined to be the McKay graph of its canonical representation.
For finite subgroups of SL, the canonical representation on C'2 is self-dual, so nij = nji for all i, j. Thus, the McKay graph of finite subgroups of SL is undirected.
In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of SL and the extended Coxeter-Dynkin diagrams of type A-D-E.
We define the Cartan matrix cV of V'' as follows:
where is the Kronecker delta.

Some results

Conjugacy Classes