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
[edit]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
[edit]- 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
[edit]- 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, …, ℓ, 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, ψℓ. 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
[edit]References
[edit]- ^ 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
- ^ McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag
Further reading
[edit]- Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7
- 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, 37, Amer. Math. Soc.: 183–186, doi:10.1090/pspum/037/604577, ISBN 9780821814406
- 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