McKay graph


In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of. Each node represents an irreducible representation of. If are irreducible representations of, then there is an arrow from to if and only if is a constituent of the tensor product Then the weight of the arrow is the number of times this constituent appears in 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 where is the Kronecker delta. A result by Robert Steinberg states that if is a representative of a conjugacy class of, then the vectors are the eigenvectors of to the eigenvalues 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 be the irreducible representations of. If
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 or sometimes as unlabeled arrows.
  • If we denote the two opposite arrows between as an undirected edge of weight. Moreover, if we omit the weight label.
We can calculate the value of using inner product on characters:
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 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:
where is the Kronecker delta.

Some results

  • If the representation is faithful, then every irreducible representation is contained in some tensor power and the McKay graph of is connected.
  • The McKay graph of a finite subgroup of has no self-loops, that is, 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
  • Felix Klein proved that the finite subgroups of are the binary polyhedral groups; all are conjugate to subgroups of 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 is generated by the matrices:
Conjugacy Classes