McKay graph

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Short description: Construction in graph theory



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 Vχi. Then the weight nij of the arrow is the number of times this constituent appears in Vχi. For finite subgroups H of GL(2,), 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 cV=(dδijnij)ij, where δ is the Kronecker delta. A result by (Steinberg 1985) states that if g is a representative of a conjugacy class of G, then the vectors ((χi(g))i are the eigenvectors of cV to the eigenvalues dχV(g), where χV is the character of the representation V.

The McKay correspondence (McKay 1982), named after John McKay, states that there is a one-to-one correspondence between the McKay graphs of the finite subgroups of SL(2,) 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 χ be its character. Let {χ1,,χd} be the irreducible representations of G. If

Vχi=jnijχj,

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 χinijχj, or sometimes as nij unlabeled arrows.
  • If nij=nji, we denote the two opposite arrows between χ i, χ j as an undirected edge of weight nij. Moreover, if nij=1, we omit the weight label.

We can calculate the value of nij using inner product , on characters:

nij=Vχi,χj=1|G|gGV(g)χi(g)χj(g).

The McKay graph of a finite subgroup of GL(2,) is defined to be the McKay graph of its canonical representation.

For finite subgroups of SL(2,), the canonical representation on 2 is self-dual, so nij=nji for all i, j. Thus, the McKay graph of finite subgroups of SL(2,) is undirected.

In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of SL(2,) and the extended Coxeter-Dynkin diagrams of type A-D-E.

We define the Cartan matrix cV of V as follows:

cV=(dδijnij)ij,

where δij is the Kronecker delta.

Some results

  • If the representation V is faithful, then every irreducible representation is contained in some tensor power Vk, and the McKay graph of V is connected.
  • The McKay graph of a finite subgroup of SL(2,) has no self-loops, that is, nii=0 for all i.
  • The arrows of the McKay graph of a finite subgroup of SL(2,) are all of weight one.

Examples

  • 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
χi×ψj1ik,1j
are the irreducible representations of A × B, where χi×ψj(a,b)=χi(a)ψj(b),(a,b)A×B. In this case, we have
(cA×cB)(χi×ψ),χn×ψp=cAχk,χncBψ,ψp.
Therefore, there is an arrow in the McKay graph of G between χi×ψj and χk×ψ 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 SL(2,) are the binary polyhedral groups; all are conjugate to subgroups of SU(2,). 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 T is generated by the SU(2,) matrices:
S=(i00i),  V=(0ii0),  U=12(εε3εε7),
where ε is a primitive eighth root of unity. In fact, we have
T={Uk,SUk,VUk,SVUkk=0,,5}.
The conjugacy classes of T are:
C1={U0=I},
C2={U3=I},
C3={±S,±V,±SV},
C4={U2,SU2,VU2,SVU2},
C5={U,SU,VU,SVU},
C6={U2,SU2,VU2,SVU2},
C7={U,SU,VU,SVU}.
The character table of T is
Conjugacy Classes C1 C2 C3 C4 C5 C6 C7
χ1 1 1 1 1 1 1 1
χ2 1 1 1 ω ω2 ω ω2
χ3 1 1 1 ω2 ω ω2 ω
χ4 3 3 1 0 0 0 0
c 2 2 0 1 1 1 1
χ5 2 2 0 ω ω2 ω ω2
χ6 2 2 0 ω2 ω ω2 ω
Here ω=e2πi/3. The canonical representation V is here denoted by c. Using the inner product, we find that the McKay graph of T is the extended Coxeter–Dynkin diagram of type E~6.

See also

References

  • Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7, https://archive.org/details/introductiontoli00jame 
  • James, Gordon; Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. 
  • Klein, Felix (1884), "Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade", Teubner (Leibniz) 
  • McKay, John (1980), "Graphs, singularities and finite groups", Proc. Symp. Pure Math., Proceedings of Symposia in Pure Mathematics (Amer. Math. Soc.) 37: 183–186, doi:10.1090/pspum/037/604577, ISBN 9780821814406 
  • McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag 
  • Riemenschneider, Oswald (2005), McKay correspondence for quotient surface singularities, Singularities in Geometry and Topology, Proceedings of the Trieste Singularity Summer School and Workshop, pp. 483–519 
  • Steinberg, Robert (1985), "Subgroups of SU2, Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics 18: 587–598, doi:10.2140/pjm.1985.118.587