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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 {\displaystyle \chi _{i},\chi _{j}} are irreducible representations of G, then there is an arrow from χ i {\displaystyle \chi _{i}} to χ j {\displaystyle \chi _{j}} if and only if χ j {\displaystyle \chi _{j}} is a constituent of the tensor product V ⊗ χ 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}}. For finite subgroups H of GL, 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 c V = i j {\displaystyle c_{V}=_{ij}} , 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 ] i {\displaystyle ]_{i}} are the eigenvectors of cV to the eigenvalues d − χ V {\displaystyle d-\chi _{V}} , where χ V {\displaystyle \chi _{V}} is the character of the representation V.
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 and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.