McKay Correspondence

In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducible representations of , then there is an arrow from to if and only if is a constituent of the tensor product $V\otimes\chi_i.$ Then the weight of the arrow is the number of times this constituent appears in $V \otimes\chi_i.$ For finite subgroups of the McKay graph of is the McKay graph of the canonical representation of .

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

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.

Definition

Let be a finite group, be a representation of and be its character. Let $\$ be the irreducible representations of . If

:$V\otimes\chi_i = \sum_j n_ \chi_j,$

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 $\chi_i\xrightarrow\chi_j,$ or sometimes as unlabeled arrows.
* If $n_ = n_,$ we denote the two opposite arrows between as an undirected edge of weight . Moreover, if $n_ = 1,$ we omit the weight label.

We can calculate the value of using inner product $\langle \cdot, \cdot \rangle$ on characters:

:$n_ = \langle V\otimes\chi_i, \chi_j\rangle = \frac\sum_ V(g)\chi_i(g)\overline.$

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 $n_=n_$ for all . 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 of as follows:

:$c_V = (d\delta_ - n_)_,$

where is the Kronecker delta.

Some results

* If the representation is faithful, then every irreducible representation is contained in some tensor power $V^,$ and the McKay graph of is connected.
* The McKay graph of a finite subgroup of has no self-loops, that is, $n_=0$ for all .
* The arrows of the McKay graph of a finite subgroup of 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 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 $\overline$ is generated by the matrices:

::$S = \left( \begin i & 0 \\ 0 & -i \end \right),\ \ V = \left( \begin 0 & i \\ i & 0 \end \right),\ \ U = \frac \left( \begin \varepsilon & \varepsilon^3 \\ \varepsilon & \varepsilon^7 \end \right),$

: where is a primitive eighth root of unity. In fact, we have

::$\overline = \.$

: The conjugacy classes of $\overline$ are:

:: $C_1 = \,$
:: $C_2 = \,$
:: $C_3 = \,$
:: $C_4 = \,$
:: $C_5 = \,$
:: $C_6 = \,$
:: $C_7 = \.$

: The character table of $\overline$ is

: Here $\omega = e^.$ The canonical representation is here denoted by . Using the inner product, we find that the McKay graph of $\overline$ is the extended Coxeter–Dynkin diagram of type $\tilde_6.$

See also

* ADE classification
* Binary tetrahedral group

References

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* {{Citation , first = Robert , last = Steinberg , title = Subgroups of $SU_2$, Dynkin diagrams and affine Coxeter elements, year = 1985 , journal = Pacific Journal of Mathematics, volume = 18 , pages = 587–598, doi = 10.2140/pjm.1985.118.587

Representation theory