McKay Quiver: Theory & Applications

• McKay quiver is a directed graph constructed from a finite group’s irreducible representations and tensor-product multiplicities, revealing links to ADE classifications.
• The quiver encodes representation theory data where arrows denote the decompositions of tensor products, forming extended Dynkin diagrams in classical cases.
• It generalizes to variants in quantum groups, group schemes, and graded settings, with applications in moduli spaces, quiver varieties, and noncommutative geometry.

Searching arXiv for recent and foundational papers on McKay quivers. A McKay quiver is a directed graph attached to a symmetry datum together with a fixed representation: in the classical formulation the datum is a finite group $G$, the vertices are irreducible representations of $G$, and the arrows record multiplicities in tensor products with a chosen $G$-module. McKay introduced the McKay quiver in 1979 for finite groups and their representations, and for finite subgroups of $SL(2,\mathbb C)$ he observed that the resulting graph is an extended Dynkin diagram of type $A$, $D$, or $E$; this observation became the starting point for a web of correspondences linking representation theory, quotient singularities, quiver varieties, and derived categories (Guo, 2022, Browne, 2020).

1. Definition and representation-theoretic construction

For a finite group $G$ with irreducible complex representations $\sigma_1,\dots,\sigma_r$ and a finite-dimensional complex representation $\rho:G\to \mathrm{GL}(V)$, one writes

$G$0

The McKay matrix is $G$1, and the McKay quiver $G$2 has one vertex $G$3 for each irreducible $G$4 and $G$5 arrows from $G$6 to $G$7 (Browne, 2020). In the finite abelian case, all irreducible representations are $G$8-dimensional, so a decomposition $G$9 makes the quiver a union of directed graphs obtained by the permutations $G$0 (Guo, 2022).

This construction has several formal features. The eigenvectors of $G$1 are exactly the columns of the character table of $G$2, and the eigenvalue corresponding to a conjugacy class $G$3 is $G$4 for any $G$5; in particular the dimension vector $G$6 is an eigenvector with eigenvalue $G$7 (Browne, 2020). Duality reverses orientation: $G$8, so $G$9 is obtained from $SL(2,\mathbb C)$0 by reversing all arrows, and $SL(2,\mathbb C)$1 is symmetric if and only if $SL(2,\mathbb C)$2 is self-dual (Browne, 2020).

An analogous definition exists for finite linearly reductive group schemes. If $SL(2,\mathbb C)$3 is a finite linearly reductive group scheme, $SL(2,\mathbb C)$4 a complete set of simple $SL(2,\mathbb C)$5-modules, and $SL(2,\mathbb C)$6 an $SL(2,\mathbb C)$7-module, the paper on domestic finite group schemes defines multiplicities by

$SL(2,\mathbb C)$8

and the McKay quiver $SL(2,\mathbb C)$9 has vertex set $A$0 and $A$1 arrows from $A$2 to $A$3 (Kirchhoff, 2015). In this formulation the combinatorial object is again the oriented graph of tensor-product multiplicities.

2. Classical correspondence and the ADE pattern

For a finite subgroup $A$4 with $A$5, the McKay quiver is an affine Dynkin quiver of type $A$6 with $A$7 (Mozgovoy, 2011). Equivalently, the McKay graph of $A$8, forgetting orientation, is an extended Dynkin diagram of ADE type, and the vertex corresponding to the trivial representation is the extending node (Craw et al., 2019). In the usual double-quiver language, each unoriented edge is replaced by a pair of opposite arrows.

The geometric content of the classical $A$9-dimensional McKay correspondence identifies nontrivial irreducible representations with irreducible components of the exceptional fiber of the crepant resolution $D$0, and the Chern classes $D$1 for $D$2 form a $D$3-basis of $D$4, dual to these components (Mozgovoy, 2011). In the preprojective formulation, if $D$5 is the affine Dynkin quiver and $D$6 its double, then

$D$7

is Morita equivalent to the skew group algebra $D$8 (Mozgovoy, 2011).

The same framed McKay quiver governs quiver-variety constructions. For $D$9, the framed McKay quiver $E$0 is obtained by adjoining a framing vertex $E$1 and a single edge between $E$2 and the trivial-representation vertex $E$3, then doubling all edges; its preprojective algebra $E$4 defines Nakajima quiver varieties $E$5 for dimension vector

$E$6

with $E$7 and $E$8 (Craw et al., 2019). For a specific non-generic stability parameter $E$9, the reduced scheme underlying $G$0 is isomorphic to the quiver variety $G$1; the same framework yields irreducibility, normality, and uniqueness of the projective symplectic resolution (Craw et al., 2019).

3. Variants and generalizations

Beyond finite groups in characteristic $G$2, McKay quivers admit several generalizations. For a finite quiver $G$3 without loops and a finite abelian group $G$4 acting admissibly, Demonet’s construction produces a generalized McKay quiver $G$5 whose vertices are pairs $G$6, where $G$7 is a vertex of the original quiver and $G$8 is an irreducible representation of the stabilizer $G$9; $\sigma_1,\dots,\sigma_r$0 is equivalent to $\sigma_1,\dots,\sigma_r$1 (Hou et al., 2011). The same paper associates to $\sigma_1,\dots,\sigma_r$2 a valued graph $\sigma_1,\dots,\sigma_r$3, proves that the positive roots of the Kac–Moody algebra $\sigma_1,\dots,\sigma_r$4 are exactly the images $\sigma_1,\dots,\sigma_r$5 for indecomposable $\sigma_1,\dots,\sigma_r$6-representations $\sigma_1,\dots,\sigma_r$7, and lifts $\sigma_1,\dots,\sigma_r$8 to $\sigma_1,\dots,\sigma_r$9 so that $\rho:G\to \mathrm{GL}(V)$0 embeds into $\rho:G\to \mathrm{GL}(V)$1 (Hou et al., 2011).

In positive characteristic, for a finite linearly reductive subgroup scheme $\rho:G\to \mathrm{GL}(V)$2, one studies the McKay quiver of $\rho:G\to \mathrm{GL}(V)$3 relative to the Frobenius-twisted $\rho:G\to \mathrm{GL}(V)$4-dimensional representation $\rho:G\to \mathrm{GL}(V)$5. As recalled in the domestic finite group scheme setting, this McKay quiver is isomorphic to one of

$\rho:G\to \mathrm{GL}(V)$6

interpreted as double quivers (Kirchhoff, 2015). A plausible implication is that the extended Dynkin pattern persists far beyond the characteristic-zero subgroup case, but now inside the representation theory of finite group schemes.

A quantum analogue also exists. Using the representation theory of $\rho:G\to \mathrm{GL}(V)$7 at a root of unity, one obtains an ADE graph $\rho:G\to \mathrm{GL}(V)$8 from a rigid commutative algebra object $\rho:G\to \mathrm{GL}(V)$9 in the fusion category, and an oriented quiver $G$00; the resulting category $G$01 is equivalent to $G$02, and its indecomposable objects give the corresponding root system (Jr et al., 2012).

A higher-dimensional graded version was introduced for arbitrary finite subgroups $G$03. In that setting, the number of arrows of degree $G$04 from $G$05 to $G$06 in the graded McKay quiver equals the multiplicity of $G$07 in $G$08, and the associated $G$09-dimensional Ginzburg dg algebra $G$10 is quasi-isomorphic to $G$11 (Liu, 2024).

4. Relations, potentials, and noncommutative geometry

In many applications the McKay quiver is used together with relations or with a potential. For $G$12, if $G$13 is the affine McKay quiver of $G$14, then the McKay quiver $G$15 of $G$16 is obtained by taking the double quiver $G$17 and adding a loop $G$18 at every vertex (Mozgovoy, 2011). The potential is

$G$19

and the Jacobian algebra $G$20 is Morita equivalent to the skew group algebra $G$21, hence provides a noncommutative crepant resolution of $G$22 (Mozgovoy, 2011). Derived equivalences identify

$G$23

This quiver-with-potential formalism is also a source of Donaldson–Thomas theory. For loop-double quivers $G$24, the universal motivic DT series is expressed in terms of Kac polynomials $G$25, and in the affine ADE McKay case one has $G$26 (Mozgovoy, 2011). Thus the same McKay quiver controls both the geometry of the crepant resolution and the motivic DT invariants of the associated $G$27-Calabi–Yau category.

For finite subgroups $G$28, Bocklandt–Schedler–Wemyss show that the skew group algebra $G$29 is graded Morita equivalent to a graded Jacobian algebra $G$30, where $G$31 is the McKay quiver and $G$32 is a homogeneous potential of degree $G$33 (Völcsey et al., 2010). In the cyclic case $G$34 with weights $G$35, the vertices are $G$36, the arrows are $G$37 from $G$38 to $G$39, and $G$40 is the signed sum of all $G$41-cycles containing $G$42 (Völcsey et al., 2010). When $G$43 for each $G$44, the corresponding potential is non-degenerate (Völcsey et al., 2010).

5. Connectivity, translation quivers, and higher representation theory

Connectivity properties of McKay quivers reflect the kernel of the chosen representation. If $G$45, then two vertices $G$46 lie in the same connected component of $G$47 if and only if the restrictions $G$48 and $G$49 are scalar multiples of each other; strong and weak connected components coincide, and the number of connected components equals the number of conjugacy classes of $G$50 contained in $G$51 (Browne, 2020). In particular, if $G$52 is faithful, then $G$53 is strongly connected (Browne, 2020). This statement supplies a precise representation-theoretic criterion for when disconnected McKay quivers appear.

In the domestic finite group scheme setting, the role of the McKay quiver is even more structural. If $G$54 is an amalgamated polyhedral group scheme and $G$55 is a Euclidean component of the stable Auslander–Reiten quiver, then for a connected component $G$56 of the separated McKay quiver one has

$G$57

as stable translation quivers (Kirchhoff, 2015). In the same setting, if $G$58 is a tube and $G$59 is the ramification index of

$G$60

then

$G$61

(Kirchhoff, 2015). Here the McKay quiver governs the Euclidean components, while ramification governs tube ranks.

McKay quivers also interact with higher Auslander–Reiten theory. If the bound quiver $G$62 of an $G$63-complete algebra is a truncation of the bound McKay quiver of a finite subgroup $G$64, then the bound quiver $G$65 of its cone is a truncation of the bound McKay quiver of a group $G$66 for some $G$67 (Zhang et al., 2016). For certain metacyclic groups embedded in $G$68 and $G$69, the corresponding McKay quivers with superpotential yield $G$70- and $G$71-representation infinite algebras; for $G$72 these examples correspond to the classical tame hereditary algebras of type $G$73 (Giovannini, 2017).

6. Moduli spaces, quiver varieties, and physical interpretations

A major application of McKay quivers is the construction of moduli spaces. For the cyclic quotient singularity $G$74, the McKay quiver has vertices $G$75, arrows

$G$76

and relations

$G$77

(Kedzierski, 2010). The paper constructs a family of McKay quiver representations on the Danilov resolution of $G$78 and proves that, for a suitable stability condition $G$79, the Danilov resolution is the normalization of the coherent component of the moduli space of $G$80-stable McKay quiver representations (Kedzierski, 2010).

For the non-Gorenstein cyclic surface singularity $G$81, the relevant object is a special McKay quiver $G$82 whose path algebra quotient is Wemyss’s reconstruction algebra; after framing and imposing stability, $G$83 appears as an irreducible connected component of a quiver variety (Bartocci et al., 2015). These quiver varieties are not of Nakajima type when $G$84 (Bartocci et al., 2015).

In gauge theory and string theory, McKay quivers describe D-brane worldvolume theories on orbifolds. For $G$85, the McKay quiver $G$86 is the double quiver of the affine ADE Dynkin diagram, and Nakajima quiver varieties built from it realize ALE spaces and instanton moduli spaces (Lechtenfeld et al., 2014). The same quiver gives the matter content and superpotential of $G$87 quiver gauge theories whose Higgs branches are these quiver varieties (Lechtenfeld et al., 2014).

Disconnected McKay quivers have a distinct physical meaning. When an orbifold group acts unfaithfully, the number of connected components is controlled by the kernel of the action, and in numerous examples each component can be interpreted through decomposition of orbifold $G$88-models; for central trivially acting subgroups, the resulting components are compatible with orbifolds with discrete torsion (Meynet et al., 2022). This suggests that connected components of a McKay quiver can carry definitive geometric meaning even when the global quiver is no longer connected.

Taken together, these developments place the McKay quiver at the center of a broad structure: it is simultaneously a tensor-product graph, a bound quiver or quiver with potential, a model for singularity and cluster categories, an organizer of Auslander–Reiten components, and a mechanism for constructing quiver varieties and moduli spaces associated with quotient singularities (Liu, 2024).