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 χ i, χ j are irreducible representations of G, then there is an arrow from χ i to χ j if and only if χ j 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
the McKay graph of H is the McKay graph of the defining 2-dimensional 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 Robert 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 χV is the character of the representation V.[1] 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.
[2] Let G be a finite group, V be a representation of G and χ be its character.
be the irreducible representations of G. If then define the McKay graph ΓG 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
is defined to be the McKay graph of its canonical representation.
For finite subgroups of
{\displaystyle n_{ij}=n_{ji}}
Thus, the McKay graph of finite subgroups of
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 cV of V as follows: where δij is the Kronecker delta.