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 ${\displaystyle V\otimes \chi _{i}.}$ Then the weight n_ij of the arrow is the number of times this constituent appears in ${\displaystyle V\otimes \chi _{i}.}$ For finite subgroups H of ⁠${\displaystyle {\text{GL}}(2,\mathbb {C} ),}$⁠ 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 c_V of the representation V of dimension d is defined by ${\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},}$ 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 ${\displaystyle ((\chi _{i}(g))_{i}}$ are the eigenvectors of c_V to the eigenvalues ${\displaystyle d-\chi _{V}(g),}$ where χ_V 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 ⁠${\displaystyle {\text{SL}}(2,\mathbb {C} )}$⁠ and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.^[2]