nLab
McKay quiver
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Graph theory
Contents
Idea
Generally, for $G$ a finite group and $V$ a linear representation of $G$ on a finite dimensional complex vector space , the McKay quiver or McKay graph associated with $V$ is the quiver whose vertices correspond to the irreducible representations $\rho_i$ of $G$ and which has $a_{i j} \in \mathbb{N}$ edges between the $i$ th and the $j$ th vertex, for $a_{i j}$ the coefficients in the expansion into irreps of the tensor product of representations of $V$ with these irreps:

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

Specifically this applies to the special case where $G \subset$ SU(2) a finite subgroup of SU(2) and $V$ its defining representation on $\mathbb{C}^2$ . 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 $\mathbb{C}^2 \sslash G$ .

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

graphics grabbed from GSV 83, p. 4

In particular, for $G =\mathbb{Z}_N \subset SU(2)$ a cyclic group of order $N$ , there are $N$ complex irreps and the McKay quiver, i.e. the extended Dynkin diagram, has $N$ -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