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

 

 are the irreducible representations of , where  In this case, we have 

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

 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:

 where  is a primitive eighth root of unity. In fact, we have

 The conjugacy classes of  are:

 
 
 
 
 
 
 

 The character table of  is

 Here  The canonical representation  is here denoted by . Using the inner product, we find that the McKay graph of  is the extended Coxeter–Dynkin diagram of type

See also 

 ADE classification
 Binary tetrahedral group

References

Further reading 
 
 
 
 
 

Representation theory