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McKay graph

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Affine (extended) Dynkin diagrams

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]

Definition

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Let G be a finite group, V be a representation of G and χ be its character. Let be the irreducible representations of G. If

then define the McKay graph ΓG of G, relative to V, as follows:

  • Each irreducible representation of G corresponds to a node in ΓG.
  • If nij > 0, there is an arrow from χ i to χ j of weight nij, written as or sometimes as nij unlabeled arrows.
  • If we denote the two opposite arrows between χ i, χ j as an undirected edge of weight nij. Moreover, if we omit the weight label.

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 the canonical representation on is self-dual, so for all i, j. 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 cV of V as follows:

where δij is the Kronecker delta.

Some results

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  • If the representation V is faithful, then every irreducible representation is contained in some tensor power and the McKay graph of V is connected.
  • The McKay graph of a finite subgroup of has no self-loops, that is, for all i.
  • The arrows of the McKay graph of a finite subgroup of are all of weight one.

Examples

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  • Suppose G = A × B, and there are canonical irreducible representations cA, cB of A, B respectively. If χ i, i = 1, …, k, are the irreducible representations of A and ψ j, j = 1, …, , are the irreducible representations of B, then
are the irreducible representations of A × B, where In this case, we have
Therefore, there is an arrow in the McKay graph of G between and if and only if there is an arrow in the McKay graph of A between χi, χk and there is an arrow in the McKay graph of B between ψ j, ψ. In this case, the weight on the arrow in the McKay graph of G is the product of the weights of the two corresponding arrows in the McKay graphs of A and B.
  • 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 ε is a primitive eighth root of unity. In fact, we have
The conjugacy classes of are:
The character table of is
Conjugacy Classes
Here The canonical representation V is here denoted by c. Using the inner product, we find that the McKay graph of is the extended Coxeter–Dynkin diagram of type

See also

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References

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  1. ^ Steinberg, Robert (1985), "Subgroups of , Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics, 18: 587–598, doi:10.2140/pjm.1985.118.587
  2. ^ McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag

Further reading

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