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In [[mathematics]], the '''McKay graph''' of a finite dimensional representation ''V'' of a finite [[group (mathematics)|group]] ''G'' is a weighted [[quiver (mathematics)|quiver]] encoding the structure of the [[representation theory]] of ''G''. Each node represents an irreducible character of ''G''. If <math>\chi_i, \chi_j</math> are irreducible representations of ''G'' then there is an arrow from <math>\chi_i</math> to <math>\chi_j</math> if and only if <math>\chi_j</math> is a constituent of the [[tensor product]] <math>V\otimes\chi_i</math>. Then the weight ''nij'' of the arrow is the number of times this constituent appears in <math>V \otimes\chi_i</math>. 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 <math> c_V = (d\delta_{ij} -n_{ij})_{ij} </math>, 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 <math> ((\chi_i(g))_i </math> are the eigenvectors of ''cV'' to the eigenvalues <math> d-\chi_V(g) </math>, where <math> \chi_V </math> is the character of the representation ''V''.

The McKay correspondence, named after [[John McKay (mathematician)|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 diagram]]s, which appear in the [[ADE classification]] of the simple [[Lie Algebra]]s.

==Definition==
Let ''G'' be a finite group, ''V'' be a [[Group representation|representation]] of ''G'' and <math> \chi </math> be its character. Let <math>\{\chi_1,\ldots,\chi_d\}</math> be the irreducible representations of ''G''. If

:<math> V\otimes\chi_i = \sum_j n_{ij} \chi_j,</math>

then define the McKay graph <math>\Gamma_G</math> of ''G'' as follow:

* To each irreducible representation of ''G'' corresponds a node in <math>\Gamma_G</math>.
* There is an arrow from <math>\chi_i</math> to <math>\chi_j</math> if and only if ''nij'' > 0 and ''nij'' is the weight of the arrow: <math>\chi_i\xrightarrow{n_{ij}}\chi_j</math>.
* If ''nij'' = ''nji'', then we put an edge between <math> \chi_i </math> and <math> \chi_j </math> 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:

:<math>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)},</math>

where <math>\langle \cdot, \cdot \rangle</math> denotes the [[inner product]] of the [[character (mathematics)|character]]s.

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:

:<math>c_V = (d\delta_{ij} - n_{ij})_{ij},</math>

where <math> \delta_{ij} </math> is the [[Kronecker delta]].

==Some results==
* 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==
*Suppose ''G'' = ''A'' × ''B'', and there are canonical irreducible representations ''cA'' and ''cB'' of ''A'' and ''B'' respectively. If <math>\chi_i</math>, ''i'' = 1, ..., ''k'', are the irreducible representations of ''A'' and <math>\psi_j</math>, ''j'' = 1, ..., l, are the irreducible representations of ''B'', then

:<math>\chi_i\times\psi_j\quad 1\leq i \leq k,\,\, 1\leq j \leq l</math>

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

:<math>\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.</math>

Therefore, there is an arrow in the McKay graph of ''G'' between <math>\chi_i\times\psi_j</math> and <math>\chi_k\times\psi_l</math> if and only if there is an arrow in the McKay graph of ''A'' between <math>\chi_i</math> and <math>\chi_k</math> and there is an arrow in the McKay graph of ''B'' between <math>\psi_j</math> and <math>\psi_l</math>. 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 <math>\overline{T}</math> 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'''):

:<math>
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),
</math>

where ε is a primitive eighth root of unity. Then, <math>\overline{T}</math> is generated by ''S'', ''U'', ''V''. In fact, we have

:<math>\overline{T} = \{U^k, SU^k,VU^k,SVU^k | k = 0,\ldots, 5\}.</math>

The conjugacy classes of <math>\overline{T}</math> are the following:

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

The character table of <math>\overline{T}</math> is

{| class="wikitable" border="1"
! Conjugacy Classes !! <math>C_1</math> !! <math>C_2</math> !! <math>C_3</math> !! <math>C_4</math> !! <math>C_5</math> !! <math>C_6</math> !! <math>C_7</math>
|-
! <math>\chi_1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|-
! <math>\chi_2</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>\omega</math>
| <math>\omega^2</math>
| <math>\omega</math>
| <math>\omega^2</math>
|-
! <math>\chi_3</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>\omega^2</math>
| <math>\omega</math>
| <math>\omega^2</math>
| <math>\omega</math>
|-
! <math>\chi_4</math>
| <math>3</math>
| <math>3</math>
| <math>-1</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
! <math>c</math>
| <math>2</math>
| <math>-2</math>
| <math>0</math>
| <math>-1</math>
| <math>-1</math>
| <math>1</math>
| <math>1</math>
|-
! <math>\chi_5</math>
| <math>2</math>
| <math>-2</math>
| <math>0</math>
| <math>-\omega</math>
| <math>-\omega^2</math>
| <math>\omega</math>
| <math>\omega^2</math>
|-
! <math>\chi_6</math>
| <math>2</math>
| <math>-2</math>
| <math>0</math>
| <math>-\omega^2</math>
| <math>-\omega</math>
| <math>\omega^2</math>
| <math>\omega</math>
|}

Here <math>\omega = e^{2\pi i/3}</math>. The canonical representation is represented by ''c''. By using the inner product, we have that the McKay graph of <math>\overline{T}</math> is the extended Coxeter-Dynkin diagram of type <math>\tilde{E}_6</math>.

== See also ==

* [[ADE classification]]
* [[Binary tetrahedral group]]

== References ==

* {{citation | title = Introduction to Lie Algebras and Representation Theory |first=James E. |last=Humphreys |publisher=Birkhäuser |year = 1972 |isbn=978-0-387-90053-7}}
* {{cite book | last1 = James | first1 = Gordon | last2 = Liebeck | first2 = Martin | title=Representations and Characters of Groups (2nd ed.) | year=2001 | publisher=Cambridge University Press | isbn=0-521-00392-X}}
* {{Citation | first = Felix | last = Klein | authorlink = Felix Klein| title = Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünten Grade | journal = Teubner | publisher = Leibniz | year = 1884}}
* {{Citation | first = John | last = McKay | authorlink = John McKay (mathematician) | title = Graphs, singularities and finite groups | journal = Proc. Symp. Pure Math. | volume = 37 | publisher = Amer. Math. Soc. | year = 1980 | pages = 183–186}}
* {{Citation | first = John | last = McKay | authorlink = John McKay (mathematician) | chapter = Representations and Coxeter Graphs |title = "The Geometric Vein", Coxeter Festschrift | year = 1982 | publisher = [[Springer-Verlag]] | location = Berlin}}
* {{Citation | first = Oswald | last = Riemenschneider | title = McKay correspondence for quotient surface singularities| year = 2005 |publisher = Singularities in Geometry and Topology, Proceedings of the Trieste Singularity Summer School and Workshop| pages = 483–519}}
* {{Citation | first = Robert | last = Steinberg | title = Subgroups of <math> SU_2 </math>, Dynkin diagrams and affine Coxeter elements| year = 1985 |journal = Pacific Journal of Mathematics| volume = 18 | pages = 587–598}}

[[Category:Representation theory]]

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