McKay graph

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Affine Dynkin diagrams.png
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 character of G. If \chi_i, \chi_j are irreducible representations of G then there is an arrow from \chi_i to \chi_j if and only if \chi_j is a constituent of the tensor product V\otimes\chi_i. Then the weight nij of the arrow is the number of times this constituent appears in V \otimes\chi_i. For finite subgroups H of GL(2, C), the McKay graph of H is the McKay graph of the canonical representation of H.

If G has n irreducible characters, then the Cartan matrix cV of the representation V of dimension d is defined by  c_V = (d\delta_{ij} -n_{ij})_{ij} , where δ is the Kronecker delta. A result by Steinberg states that if g is a representative of a conjugacy class of G, then the vectors  ((\chi_i(g))_i are the eigenvectors of cV to the eigenvalues  d-\chi_V(g) , where  \chi_V is the character of the representation V.

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 SL(2, C) and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie Algebras.

Definition[edit]

Let G be a finite group, V be a representation of G and  \chi be its character. Let \{\chi_1,\ldots,\chi_d\} be the irreducible representations of G. If

	V\otimes\chi_i = \sum_j n_{ij} \chi_j,

then define the McKay graph \Gamma_G of G as follow:

  • To each irreducible representation of G corresponds a node in \Gamma_G.
  • There is an arrow from \chi_i to \chi_j if and only if nij > 0 and nij is the weight of the arrow: \chi_i\xrightarrow{n_{ij}}\chi_j.
  • If nij = nji, then we put an edge between  \chi_i and  \chi_j instead of a double arrow. Moreover, if nij = 1, then we do not write the weight of the corresponding arrow.

We can calculate the value of nij by considering the inner product. We have the following formula:

n_{ij} = \langle V\otimes\chi_i, \chi_j\rangle = \frac{1}{|G|}\sum_{g\in G} V(g)\chi_i(g)\overline{\chi_j(g)},

where \langle \cdot, \cdot \rangle denotes the inner product of the characters.

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

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

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

We define the Cartan matrix cV of V as follow:

c_V = (d\delta_{ij} - n_{ij})_{ij},

where  \delta_{ij} is the Kronecker delta.

Some results[edit]

  • If the representation V of a finite group G is faithful, then the McKay graph of V is connected.
  • The McKay graph of a finite subgroup of SL(2, C) has no self-loops, that is, nii = 0 for all i.
  • The weights of the arrows of the McKay graph of a finite subgroup of SL(2, C) are always less or equal than one.

Examples[edit]

  • Suppose G = A × B, and there are canonical irreducible representations cA and cB of A and B respectively. If \chi_i, i = 1, ..., k, are the irreducible representations of A and \psi_j, j = 1, ..., l, are the irreducible representations of B, then
\chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq l

are the irreducible representations of A\times B, where \chi_i\times\psi_j(a,b) = \chi_i(a)\psi_j(b), (a,b)\in A\times B. In this case, we have

\langle (c_A\times c_B)\otimes (\chi_i\times\psi_l), \chi_n\times\psi_p\rangle = \langle c_A\otimes \chi_k, \chi_n\rangle\cdot \langle c_B\otimes \psi_l, \psi_p\rangle.

Therefore, there is an arrow in the McKay graph of G between \chi_i\times\psi_j and \chi_k\times\psi_l if and only if there is an arrow in the McKay graph of A between \chi_i and \chi_k and there is an arrow in the McKay graph of B between \psi_j and \psi_l. 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, C) are the binary polyhedral groups. 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, let \overline{T} be the binary tetrahedral group. Every finite subgroup of SL(2, C) is conjugate to a finite subgroup of SU(2, C). Consider the matrices in SU(2, C):

S = \left( \begin{array}{cc}
i & 0 \\
0 & -i \end{array} \right) , 
V = \left( \begin{array}{cc}
0 & i \\
i & 0 \end{array} \right), 
U = \frac{1}{\sqrt{2}} \left( \begin{array}{cc}
\epsilon & \epsilon^3 \\
\epsilon & \epsilon^7 \end{array} \right),

where ε is a primitive eighth root of unity. Then, \overline{T} is generated by S, U, V. In fact, we have

\overline{T} = \{U^k, SU^k,VU^k,SVU^k | k = 0,\ldots, 5\}.

The conjugacy classes of \overline{T} are the following:

C_1 = \{U^0 = I\},
C_2 = \{U^3 = - I\},
C_3 = \{\pm S, \pm V, \pm SV\},
C_4 = \{U^2, SU^2, VU^2, SVU^2\},
C_5 = \{-U, SU, VU, SVU\},
C_6 = \{-U^2, -SU^2, -VU^2, -SVU^2\},
C_7 = \{U, -SU, -VU, -SVU\}.

The character table of \overline{T} is

Conjugacy Classes C_1 C_2 C_3 C_4 C_5 C_6 C_7
\chi_1 1 1 1 1 1 1 1
\chi_2 1 1 1 \omega \omega^2 \omega \omega^2
\chi_3 1 1 1 \omega^2 \omega \omega^2 \omega
\chi_4 3 3 -1 0 0 0 0
c 2 -2 0 -1 -1 1 1
\chi_5 2 -2 0 -\omega -\omega^2 \omega \omega^2
\chi_6 2 -2 0 -\omega^2 -\omega \omega^2 \omega

Here \omega = e^{2\pi i/3}. The canonical representation is represented by c. By using the inner product, we have that the McKay graph of \overline{T} is the extended Coxeter-Dynkin diagram of type \tilde{E}_6.

See also[edit]

References[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 fünten Grade", Teubner (Leibniz) 
  • McKay, John (1980), "Graphs, singularities and finite groups", Proc. Symp. Pure Math. (Amer. Math. Soc.) 37: 183–186, doi:10.1090/pspum/037/604577 
  • 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  SU_2 , Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics 18: 587–598