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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 V ⊗ χ i . {\displaystyle V\otimes \chi _{i}.} {\displaystyle V\otimes \chi _{i}.} Then the weight nij of the arrow is the number of times this constituent appears in V ⊗ χ i . {\displaystyle V\otimes \chi _{i}.} {\displaystyle V\otimes \chi _{i}.} For finite subgroups H of ⁠ GL ( 2 , C ) , {\displaystyle {\text{GL}}(2,\mathbb {C} ),} {\displaystyle {\text{GL}}(2,\mathbb {C} ),}⁠ 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 c V = ( d δ i j − n i j ) i j , {\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},} {\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},} 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 ( ( χ i ( g ) ) i {\displaystyle ((\chi _{i}(g))_{i}} {\displaystyle ((\chi _{i}(g))_{i}} are the eigenvectors of cV to the eigenvalues d − χ V ( g ) , {\displaystyle d-\chi _{V}(g),} {\displaystyle d-\chi _{V}(g),} 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 ⁠ SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} {\displaystyle {\text{SL}}(2,\mathbb {C} )}⁠ and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.[2]

Definition

Let G be a finite group, V be a representation of G and χ be its character. Let { χ 1 , … , χ d } {\displaystyle \{\chi _{1},\ldots ,\chi _{d}\}} {\displaystyle \{\chi _{1},\ldots ,\chi _{d}\}} be the irreducible representations of G. If

V ⊗ χ i = ∑ j n i j χ j , {\displaystyle V\otimes \chi _{i}=\sum \nolimits _{j}n_{ij}\chi _{j},} {\displaystyle V\otimes \chi _{i}=\sum \nolimits _{j}n_{ij}\chi _{j},}

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 χ i → n i j χ j , {\displaystyle \chi _{i}\xrightarrow {n_{ij}} \chi _{j},} {\displaystyle \chi _{i}\xrightarrow {n_{ij}} \chi _{j},} or sometimes as nij unlabeled arrows.
  • If n i j = n j i , {\displaystyle n_{ij}=n_{ji},} {\displaystyle n_{ij}=n_{ji},} we denote the two opposite arrows between χ i, χ j as an undirected edge of weight nij. Moreover, if n i j = 1 , {\displaystyle n_{ij}=1,} {\displaystyle n_{ij}=1,} we omit the weight label.

We can calculate the value of nij using inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } {\displaystyle \langle \cdot ,\cdot \rangle } on characters:

n i j = ⟨ V ⊗ χ i , χ j ⟩ = 1 | G | ∑ g ∈ G V ( g ) χ i ( g ) χ j ( g ) ¯ . {\displaystyle 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)}}.} {\displaystyle 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)}}.}

The McKay graph of a finite subgroup of ⁠ GL ( 2 , C ) {\displaystyle {\text{GL}}(2,\mathbb {C} )} {\displaystyle {\text{GL}}(2,\mathbb {C} )}⁠ is defined to be the McKay graph of its canonical representation.

For finite subgroups of ⁠ SL ( 2 , C ) , {\displaystyle {\text{SL}}(2,\mathbb {C} ),} {\displaystyle {\text{SL}}(2,\mathbb {C} ),}⁠ the canonical representation on ⁠ C 2 {\displaystyle \mathbb {C} ^{2}} {\displaystyle \mathbb {C} ^{2}}⁠ is self-dual, so n i j = n j i {\displaystyle n_{ij}=n_{ji}} {\displaystyle n_{ij}=n_{ji}} for all i, j. Thus, the McKay graph of finite subgroups of ⁠ SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} {\displaystyle {\text{SL}}(2,\mathbb {C} )}⁠ is undirected.

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

We define the Cartan matrix cV of V as follows:

c V = ( d δ i j − n i j ) i j , {\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},} {\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},}

where δij is the Kronecker delta.

Some results

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

Examples

  • 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
χ i × ψ j 1 ≤ i ≤ k , 1 ≤ j ≤ ℓ {\displaystyle \chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell } {\displaystyle \chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell }
are the irreducible representations of A × B, where χ i × ψ j ( a , b ) = χ i ( a ) ψ j ( b ) , ( a , b ) ∈ A × B . {\displaystyle \chi _{i}\times \psi _{j}(a,b)=\chi _{i}(a)\psi _{j}(b),(a,b)\in A\times B.} {\displaystyle \chi _{i}\times \psi _{j}(a,b)=\chi _{i}(a)\psi _{j}(b),(a,b)\in A\times B.} In this case, we have
⟨ ( c A × c B ) ⊗ ( χ i × ψ ℓ ) , χ n × ψ p ⟩ = ⟨ c A ⊗ χ k , χ n ⟩ ⋅ ⟨ c B ⊗ ψ ℓ , ψ p ⟩ . {\displaystyle \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 .} {\displaystyle \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 G between χ i × ψ j {\displaystyle \chi _{i}\times \psi _{j}} {\displaystyle \chi _{i}\times \psi _{j}} and χ k × ψ ℓ {\displaystyle \chi _{k}\times \psi _{\ell }} {\displaystyle \chi _{k}\times \psi _{\ell }} 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 ⁠ SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} {\displaystyle {\text{SL}}(2,\mathbb {C} )}⁠ are the binary polyhedral groups; all are conjugate to subgroups of ⁠ SU ( 2 , C ) . {\displaystyle {\text{SU}}(2,\mathbb {C} ).} {\displaystyle {\text{SU}}(2,\mathbb {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 T ¯ {\displaystyle {\overline {T}}} {\displaystyle {\overline {T}}} is generated by the ⁠ SU ( 2 , C ) {\displaystyle {\text{SU}}(2,\mathbb {C} )} {\displaystyle {\text{SU}}(2,\mathbb {C} )}⁠ matrices:
S = ( i 0 0 − i ) ,     V = ( 0 i i 0 ) ,     U = 1 2 ( ε ε 3 ε ε 7 ) , {\displaystyle 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),} {\displaystyle 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 ε is a primitive eighth root of unity. In fact, we have
T ¯ = { U k , S U k , V U k , S V U k ∣ k = 0 , … , 5 } . {\displaystyle {\overline {T}}=\{U^{k},SU^{k},VU^{k},SVU^{k}\mid k=0,\ldots ,5\}.} {\displaystyle {\overline {T}}=\{U^{k},SU^{k},VU^{k},SVU^{k}\mid k=0,\ldots ,5\}.}
The conjugacy classes of T ¯ {\displaystyle {\overline {T}}} {\displaystyle {\overline {T}}} are:
C 1 = { U 0 = I } , {\displaystyle C_{1}=\{U^{0}=I\},} {\displaystyle C_{1}=\{U^{0}=I\},}
C 2 = { U 3 = − I } , {\displaystyle C_{2}=\{U^{3}=-I\},} {\displaystyle C_{2}=\{U^{3}=-I\},}
C 3 = { ± S , ± V , ± S V } , {\displaystyle C_{3}=\{\pm S,\pm V,\pm SV\},} {\displaystyle C_{3}=\{\pm S,\pm V,\pm SV\},}
C 4 = { U 2 , S U 2 , V U 2 , S V U 2 } , {\displaystyle C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},} {\displaystyle C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},}
C 5 = { − U , S U , V U , S V U } , {\displaystyle C_{5}=\{-U,SU,VU,SVU\},} {\displaystyle C_{5}=\{-U,SU,VU,SVU\},}
C 6 = { − U 2 , − S U 2 , − V U 2 , − S V U 2 } , {\displaystyle C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},} {\displaystyle C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},}
C 7 = { U , − S U , − V U , − S V U } . {\displaystyle C_{7}=\{U,-SU,-VU,-SVU\}.} {\displaystyle C_{7}=\{U,-SU,-VU,-SVU\}.}
The character table of T ¯ {\displaystyle {\overline {T}}} {\displaystyle {\overline {T}}} is
Conjugacy Classes C 1 {\displaystyle C_{1}} {\displaystyle C_{1}} C 2 {\displaystyle C_{2}} {\displaystyle C_{2}} C 3 {\displaystyle C_{3}} {\displaystyle C_{3}} C 4 {\displaystyle C_{4}} {\displaystyle C_{4}} C 5 {\displaystyle C_{5}} {\displaystyle C_{5}} C 6 {\displaystyle C_{6}} {\displaystyle C_{6}} C 7 {\displaystyle C_{7}} {\displaystyle C_{7}}
χ 1 {\displaystyle \chi _{1}} {\displaystyle \chi _{1}} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1}
χ 2 {\displaystyle \chi _{2}} {\displaystyle \chi _{2}} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} ω {\displaystyle \omega } {\displaystyle \omega } ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}} ω {\displaystyle \omega } {\displaystyle \omega } ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}}
χ 3 {\displaystyle \chi _{3}} {\displaystyle \chi _{3}} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1} ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}} ω {\displaystyle \omega } {\displaystyle \omega } ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}} ω {\displaystyle \omega } {\displaystyle \omega }
χ 4 {\displaystyle \chi _{4}} {\displaystyle \chi _{4}} 3 {\displaystyle 3} {\displaystyle 3} 3 {\displaystyle 3} {\displaystyle 3} − 1 {\displaystyle -1} {\displaystyle -1} 0 {\displaystyle 0} {\displaystyle 0} 0 {\displaystyle 0} {\displaystyle 0} 0 {\displaystyle 0} {\displaystyle 0} 0 {\displaystyle 0} {\displaystyle 0}
c {\displaystyle c} {\displaystyle c} 2 {\displaystyle 2} {\displaystyle 2} − 2 {\displaystyle -2} {\displaystyle -2} 0 {\displaystyle 0} {\displaystyle 0} − 1 {\displaystyle -1} {\displaystyle -1} − 1 {\displaystyle -1} {\displaystyle -1} 1 {\displaystyle 1} {\displaystyle 1} 1 {\displaystyle 1} {\displaystyle 1}
χ 5 {\displaystyle \chi _{5}} {\displaystyle \chi _{5}} 2 {\displaystyle 2} {\displaystyle 2} − 2 {\displaystyle -2} {\displaystyle -2} 0 {\displaystyle 0} {\displaystyle 0} − ω {\displaystyle -\omega } {\displaystyle -\omega } − ω 2 {\displaystyle -\omega ^{2}} {\displaystyle -\omega ^{2}} ω {\displaystyle \omega } {\displaystyle \omega } ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}}
χ 6 {\displaystyle \chi _{6}} {\displaystyle \chi _{6}} 2 {\displaystyle 2} {\displaystyle 2} − 2 {\displaystyle -2} {\displaystyle -2} 0 {\displaystyle 0} {\displaystyle 0} − ω 2 {\displaystyle -\omega ^{2}} {\displaystyle -\omega ^{2}} − ω {\displaystyle -\omega } {\displaystyle -\omega } ω 2 {\displaystyle \omega ^{2}} {\displaystyle \omega ^{2}} ω {\displaystyle \omega } {\displaystyle \omega }
Here ω = e 2 π i / 3 . {\displaystyle \omega =e^{2\pi i/3}.} {\displaystyle \omega =e^{2\pi i/3}.} The canonical representation V is here denoted by c. Using the inner product, we find that the McKay graph of T ¯ {\displaystyle {\overline {T}}} {\displaystyle {\overline {T}}} is the extended Coxeter–Dynkin diagram of type E ~ 6 . {\displaystyle {\tilde {E}}_{6}.} {\displaystyle {\tilde {E}}_{6}.}

See also

References

  1. Steinberg, Robert (1985), "Subgroups of S U 2 {\displaystyle SU_{2}} {\displaystyle SU_{2}}, Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics, 18: 587–598, doi:10.2140/pjm.1985.118.587
  2. McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag

Further reading