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McKay quiver

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Contents

Idea

Generally, for GG a finite group and VV a linear representation of GG on a finite dimensional complex vector space, the McKay quiver or McKay graph associated with VV is the quiver whose vertices correspond to the irreducible representations ρ i\rho_i of GG and which has a ija_{i j} \in \mathbb{N} edges between the iith and the jjth vertex, for a ija_{i j} the coefficients in the expansion into irreps of the tensor product of representations of VV with these irreps:

Vρ ija ijρ j. V \otimes \rho_i \;\simeq\; \underset{j}{\bigoplus} a_{i j} \cdot \rho_j \,.

Specifically this applies to the special case where GG \subset SU(2) a finite subgroup of SU(2) and VV its defining representation on 2\mathbb{C}^2. The McKay correspondence states that in this case the corresponding McKay quivers are Dynkin quivers/Dynkin diagrams in the same ADE classification as the ADE singularity 2G\mathbb{C}^2 \sslash G.

More precisely: If one uses all irreducible representations including the 1-dimensiona trivial representation ρ 0\rho_0 then one gets the “extended Dynkin diagram”, where the extra node corresponds to ρ 0\rho_0. This is the vertex indicated by a cross in the following diagrams:

graphics grabbed from GSV 83, p. 4

In particular, for G= NSU(2)G =\mathbb{Z}_N \subset SU(2) a cyclic group of order NN, there are NN complex irreps and the McKay quiver, i.e. the extended Dynkin diagram, has NN-vertices, connected by edges to form a circle.

References

The construction is due to

  • John McKay, Graphs, singularities, and finite groups Proc. Symp. Pure Math. Vol. 37. No. 183. 1980

Interpretation in terms of equivariant K-theory of the corresponding ADE-singularity and plain K-theory of its blowup is due to

See also

Last revised on January 23, 2019 at 05:32:39. See the history of this page for a list of all contributions to it.