Construction in graph theory
![](//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Affine_Dynkin_diagrams.png/320px-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 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
Then the weight nij of the arrow is the number of times this constituent appears in
For finite subgroups H of
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
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
are the eigenvectors of cV to the eigenvalues
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
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
be the irreducible representations of G. If
![{\displaystyle V\otimes \chi _{i}=\sum _{j}n_{ij}\chi _{j},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53caadad4bce4afb7a8eaa0b1107f120ef10c784)
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
or sometimes as nij unlabeled arrows. - If
we denote the two opposite arrows between χ i, χ j as an undirected edge of weight nij. Moreover, if
we omit the weight label.
We can calculate the value of nij using inner product
on characters:
![{\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)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5cd3411128256a18aa332d737f42dbeb680b24)
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
for all i, j. 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 cV of V as follows:
![{\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1b28a9198e066ba980aea95dcffa49ed79f825)
where δij is the Kronecker delta.
Some results
- If the representation V is faithful, then every irreducible representation is contained in some tensor power
and the McKay graph of V is connected. - The McKay graph of a finite subgroup of
has no self-loops, that is,
for all i. - The arrows of the McKay graph of a finite subgroup of
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
![{\displaystyle \chi _{i}\times \psi _{j}\quad 1\leq i\leq k,\,\,1\leq j\leq \ell }](https://wikimedia.org/api/rest_v1/media/math/render/svg/91c75c2c43f300ef2c320191601319f0540f54e0)
- are the irreducible representations of A × B, where
In this case, we have
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0051492b14aaf43a95159e1b6180be9a81a8ffcb)
- Therefore, there is an arrow in the McKay graph of G between
and
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
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
is generated by the
matrices:
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1256bd310324ef56564461419a4260838d02316)
- where ε is a primitive eighth root of unity. In fact, we have
![{\displaystyle {\overline {T}}=\{U^{k},SU^{k},VU^{k},SVU^{k}\mid k=0,\ldots ,5\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f775287c0cac0b76b4c5f4369a57f4f17e3a1cc1)
- The conjugacy classes of
are:
![{\displaystyle C_{1}=\{U^{0}=I\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/919a45c239967d19a629b2e786e0a842b9950944)
![{\displaystyle C_{2}=\{U^{3}=-I\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97baaf7089868d96669594bfa6ac5e57efcadb66)
![{\displaystyle C_{3}=\{\pm S,\pm V,\pm SV\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ba75c4e8af8ae780c2da2fb8a62c6865e1e795)
![{\displaystyle C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af2639db9a94778dd86bdf02c9b75a399a8af36a)
![{\displaystyle C_{5}=\{-U,SU,VU,SVU\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f08d9f8ff30ba0848df87d85850bacffdd0441e0)
![{\displaystyle C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd49bade31f3d3528ea5e02315ebaca4c6be7f9)
![{\displaystyle C_{7}=\{U,-SU,-VU,-SVU\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bf19b1ce93feb32281a2c20d303d0d857e8241)
- The character table of
is
Conjugacy Classes | ![{\displaystyle C_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/babf569931f1a7b5182b9bec51873c2f5692fbb8) | ![{\displaystyle C_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec545f7870665e1028b7492746848d149878808) | ![{\displaystyle C_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9abeb5057b7afbf88e3169101849354f13c65) | ![{\displaystyle C_{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59cf1a8e46030a81fc175be95561e4f161241a70) | ![{\displaystyle C_{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d93b495ef896a93e1183aa55f53d32daa538738) | ![{\displaystyle C_{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17692472a5b1f5de8b8b27603154198c16b61fbc) | |
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- Here
The canonical representation V is here denoted by c. Using the inner product, we find that the McKay graph of
is the extended Coxeter–Dynkin diagram of type ![{\displaystyle {\tilde {E}}_{6}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/754827ff2b2d00b950649537cb33b0ad81046b17)
See also
- ADE classification
- Binary tetrahedral group
References
- ^ Steinberg, Robert (1985), "Subgroups of
, Dynkin diagrams and affine Coxeter elements", Pacific Journal of Mathematics, 18: 587–598, doi:10.2140/pjm.1985.118.587 - ^ McKay, John (1982), "Representations and Coxeter Graphs", "The Geometric Vein", Coxeter Festschrift, Berlin: Springer-Verlag
Further reading
- 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 fünften Grade", Teubner, Leibniz
- McKay, John (1980), "Graphs, singularities and finite groups", Proc. Symp. Pure Math., Proceedings of Symposia in Pure Mathematics, 37, Amer. Math. Soc.: 183–186, doi:10.1090/pspum/037/604577, ISBN 9780821814406
- 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