McKay Graph¶

Encode tensoring by a fixed finite-group representation as a multiplicity-weighted quiver on irreducible representations, turning fusion rules into adjacency, paths, spectra, and—in the SU(2) case—affine ADE structure.

Version: v1 · 2026-08-30 · History
Domain-specific #: 2251
Origin domain: mathematics
Subdomain: representation theory
Aliases: McKay quiver

Core Idea¶

The McKay graph of a finite group \(G\) relative to a fixed finite-dimensional complex representation \(V\) is the multiplication table for tensoring by \(V\), rendered as a quiver. Its vertices are the isomorphism classes of irreducible complex representations \(\rho_0,\ldots,\rho_r\) of \(G\). Decompose

\[ V\otimes\rho_i\cong\bigoplus_{j=0}^{r} a_{ij}\rho_j, \qquad a_{ij}\in\mathbb Z_{\geq0}. \]

Under the convention used here, the quiver has \(a_{ij}\) arrows from vertex \(i\) to vertex \(j\), or one arrow labeled by weight \(a_{ij}\). The resulting adjacency matrix \(A=(a_{ij})\) is therefore not an arbitrary connection matrix: its \(i\)-th row is exactly the irreducible decomposition of \(V\otimes\rho_i\). This is McKay's “representation graph” construction in its modern directed, multiplicity-preserving form.

Scope of Application¶

The abstraction's home is finite-group representation theory and the fields that use its tensoring data.

Representation-ring calculation: store multiplication by a chosen representation as a nonnegative integer matrix and obtain repeated tensor-power multiplicities from matrix powers.
Character-theoretic computation: recover entries by irreducible decomposition or character inner products, and read spectral information from character-table columns.
Faithfulness and generation: use reachability from the trivial vertex to determine which irreducibles occur in tensor powers; for the classical complex construction, faithful \(V\) yields a connected graph.
Finite subgroups of \(\mathrm{SU}(2)\): identify cyclic and binary polyhedral groups with affine ADE diagrams through the defining two-dimensional representation.
Kleinian singularities and invariant theory: connect the affine diagram from representation theory with the exceptional-curve diagram in the minimal resolution of \(\mathbb C^2/G\); the extending vertex corresponds to the trivial representation.
Quiver and preprojective algebra constructions: use the directed graph and path algebra as combinatorial input, with additional relations when reconstructing skew-group or preprojective structures.
Generalized settings: analogous McKay matrices occur for compact groups, finite-dimensional Hopf algebras, positive characteristic, and higher-dimensional quotient actions, but semisimplicity, vertex choice, and constituent notions must be restated rather than silently imported.

Clarity¶

Three labels prevent most mistakes: McKay graph of which group, relative to which representation, under which arrow convention? Writing \(\Gamma(G,V)\) is better than “the McKay graph of \(G\)” unless a canonical representation has already been fixed. For a subgroup \(G\subset\mathrm{SU}(2)\), convention often supplies the defining two-dimensional \(V\); outside that context it does not.

Manages Complexity¶

The graph turns a family of decomposition equations into one reusable operator. Without it, each new tensor power seems to require a fresh Clebsch–Gordan calculation. With it, iteration becomes matrix multiplication, reachability becomes occurrence in some tensor power, and recurrence relations for multiplicities become walk recurrences.

Abstract Reasoning¶

Let \(R(G)\) be the complex representation ring, with irreducible basis \([\rho_0],\ldots,[\rho_r]\). Tensoring by \(V\) is the positive integral linear operator

\[ T_V:R(G)\to R(G),\qquad x\mapsto[V]x. \]

The McKay matrix is simply the matrix of \(T_V\) in that basis. This explains why the construction simultaneously behaves like algebra and graph theory: direct-sum coefficients become edge multiplicities, composition of tensoring operators becomes matrix multiplication, and repeated application becomes weighted walks.

Knowledge Transfer¶

The full instrument transfers literally across finite groups and choices of complex representation. The recipe remains stable: irreducibles become vertices, tensor constituents become directed neighbors, multiplicities become weights, and matrix powers count iterated tensoring. What changes is the group, the irreducible roster, and the operator \([V]\).

Knowledge learned in one example travels productively. A directed cycle suggests tensoring by a one-dimensional character that permutes irreducibles. Symmetric adjacency suggests checking self-duality. Multiple components suggest checking the kernel of \(V\) and the effective quotient.

Relationships to Other Abstractions¶

Current abstraction McKay Graph Domain-specific

Parents (3) — more general patterns this builds on

McKay Graph is a kind of Network Prime

McKay Graph instantiates Representation because it maps a tensoring operator into a graph medium with an exact readback.
McKay Graph is a kind of Representation Prime

McKay Graph instantiates Representation because it maps a tensoring operator into a graph medium with an exact readback.
McKay Graph presupposes Decomposition Prime

McKay Graph instantiates Representation because it maps a tensoring operator into a graph medium with an exact readback.

Hierarchy paths (3) — routes to 3 parentless roots

McKay Graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Show alternative paths (2)

Neighborhood in Abstraction Space¶

McKay Graph sits in a sparse region of the domain-specific corpus (66^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Categorical Algebra & Model Systems (8 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08