McKay graph

Let be a finite group, be a representation of and be its character. Let \{\chi_1,\ldots,\chi_d\} be the irreducible representations of . If :V\otimes\chi_i = \sum\nolimits_j n_{ij} \chi_j, then define the McKay graph of , relative to , as follows: • Each irreducible representation of corresponds to a node in . • If , there is an arrow from to of weight , written as \chi_i\xrightarrow{n_{ij}}\chi_j, or sometimes as unlabeled arrows. • If n_{ij} = n_{ji}, we denote the two opposite arrows between as an undirected edge of weight . Moreover, if n_{ij} = 1, we omit the weight label. We can calculate the value of using inner product \langle \cdot, \cdot \rangle on characters: :n_{ij} = \langle V\otimes\chi_i, \chi_j\rangle = \frac{1}\sum_{g\in G} V(g)\chi_i(g)\overline{\chi_j(g)}. The McKay graph of a finite subgroup of {{tmath|\text{GL}(2, \C)}} is defined to be the McKay graph of its canonical representation. For finite subgroups of {{tmath|\text{SL}(2, \C),}} the canonical representation on is self-dual, so n_{ij}=n_{ji} for all . Thus, the McKay graph of finite subgroups of {{tmath|\text{SL}(2, \C)}} is undirected. In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of {{tmath|\text{SL}(2, \C)}} and the extended Coxeter-Dynkin diagrams of type A-D-E. We define the Cartan matrix of as follows: :c_V = (d\delta_{ij} - n_{ij})_{ij}, where is the Kronecker delta. ==Some results==

• If the representation is faithful, then every irreducible representation is contained in some tensor power V^{\otimes k}, and the McKay graph of is connected. • The McKay graph of a finite subgroup of {{tmath|\text{SL}(2, \C)}} has no self-loops, that is, n_{ii}=0 for all . • The arrows of the McKay graph of a finite subgroup of {{tmath|\text{SL}(2, \C)}} are all of weight one. ==Examples==

• Suppose , and there are canonical irreducible representations of respectively. If , are the irreducible representations of and , are the irreducible representations of , then :: \chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq \ell : are the irreducible representations of , 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_\ell), \chi_n\times\psi_p\rangle = \langle c_A\otimes \chi_k, \chi_n\rangle\cdot \langle c_B\otimes \psi_\ell, \psi_p\rangle. : Therefore, there is an arrow in the McKay graph of between \chi_i\times\psi_j and \chi_k\times\psi_\ell if and only if there is an arrow in the McKay graph of between and there is an arrow in the McKay graph of between . In this case, the weight on the arrow in the McKay graph of is the product of the weights of the two corresponding arrows in the McKay graphs of and . • Felix Klein proved that the finite subgroups of {{tmath|\text{SL}(2, \C)}} are the binary polyhedral groups; all are conjugate to subgroups of {{tmath|\text{SU}(2, \C).}} 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 \overline{T} is generated by the {{tmath|\text{SU}(2, \C)}} matrices: :: 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} \varepsilon & \varepsilon^3 \\ \varepsilon & \varepsilon^7 \end{array} \right), : where is a primitive eighth root of unity. In fact, we have ::\overline{T} = \{U^k, SU^k,VU^k,SVU^k \mid k = 0,\ldots, 5\}. : The conjugacy classes of \overline{T} are: :: 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 : Here \omega = e^{2\pi i/3}. The canonical representation is here denoted by . Using the inner product, we find that the McKay graph of \overline{T} is the extended Coxeter–Dynkin diagram of type \tilde{E}_6. == See also ==