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.^[1]^[2]

The choice of \(V\) is part of the identity. The same group can have many McKay graphs, because different tensoring representations define different linear operators on its representation ring. Direction is also a convention that must be declared. Some sources reverse all arrows or place \(a_{ij}\) in the transpose position. Nothing mathematical changes if the reversal is global and explicit, but mixing conventions corrupts path and matrix calculations.

The graph compresses repeated tensor decomposition. If \(A\) uses the row convention above, then

\[ (A^k)_{ij}=[V^{\otimes k}\otimes\rho_i:\rho_j], \]

the multiplicity of \(\rho_j\) in the \(k\)-fold tensor product. Thus weighted paths of length \(k\) count tensor-power multiplicities. The dimension vector \(d=(\dim\rho_i)_i\) satisfies

\[ A d=(\dim V)d, \]

because dimensions agree on both sides of every decomposition. Character evaluation also diagonalizes the tensoring operator: Steinberg showed that the columns of the character table give eigenvectors of \((\dim V)I-A\), with eigenvalues \(\dim V-\chi_V(g)\) for conjugacy-class representatives \(g\).^[3]

The celebrated ADE phenomenon is a special case, not the definition. When \(G\) is a finite subgroup of \(\mathrm{SU}(2)\) and \(V\) is the defining two-dimensional representation, \(V\) is self-dual, so arrows occur in opposite pairs. Apart from the trivial and order-two cyclic degeneracies, collapsing each pair gives an undirected affine simply laced Dynkin diagram of type \(\widetilde A\), \(\widetilde D\), or \(\widetilde E\); the trivial representation marks the extending vertex.^[1]^[3]^[2] For \(G=\{\pm I\}\), the defining representation restricts as two copies of the nontrivial character, producing the affine \(\widetilde A_1\) double-edge, or multiplicity-two, case. If the trivial subgroup is admitted, its sole vertex carries two loops. General McKay graphs can be directed, disconnected, have loops, and have multiplicities greater than one. Treating ADE features as universal would erase the construction's actual scope.

The structural signature is therefore not “a graph associated with a group.” It is the exact representation-theoretic operator

\[ [\rho_i]\longmapsto [V][\rho_i] \]

encoded as nonnegative adjacency data over the irreducible basis of the complex representation ring.

Structural Signature¶

Sig role-phrases:

the finite group and coefficient setting — a declared finite group \(G\) and a semisimple representation category, classically finite-dimensional complex representations
the complete irreducible roster — one vertex for each isomorphism class \(\rho_i\), including the trivial representation \(\rho_0\)
the fixed tensoring representation — the chosen \(G\)-representation \(V\) whose multiplication action the graph encodes
the tensor-product decompositions — the exact direct-sum expansions \(V\otimes\rho_i\cong\bigoplus_j a_{ij}\rho_j\)
the direction convention — arrows \(i\to j\) mean that \(\rho_j\) occurs in \(V\otimes\rho_i\), unless a globally reversed convention is explicitly chosen
the multiplicity-preserving edges — parallel arrows or integer weights retain every coefficient \(a_{ij}\), including loops and asymmetric pairs when present
the adjacency or McKay matrix — the nonnegative integer matrix \(A=(a_{ij})\) representing multiplication by \([V]\) in the irreducible basis
the iterated-walk interpretation — matrix powers and weighted walks recover multiplicities in \(V^{\otimes k}\otimes\rho_i\)
the derived structural tests — dimension, character-spectrum, self-duality, faithfulness/connectivity, and special ADE consequences used only under their proper hypotheses

The locked signature is

\[ (G,\operatorname{Irr}(G),V) +\text{exact tensor decompositions} +\text{declared orientation} \Longleftrightarrow \text{multiplicity adjacency operator }A. \]

Recognition test: choose a vertex \(i\), read its outgoing weighted edges, and reconstruct \(\bigoplus_j a_{ij}\rho_j\). If this is not isomorphic to \(V\otimes\rho_i\) for every vertex, if an irreducible is missing from the roster, or if multiplicities have been collapsed, the artifact is not a valid McKay graph for the declared pair \((G,V)\).

What It Is Not¶

Not the ordinary Cayley graph of a group. Cayley vertices are group elements and edges multiply by generators; McKay vertices are irreducible representations and edges tensor by a representation.
Not the character table. The character table evaluates irreducible characters on conjugacy classes. It can compute and spectrally analyze a McKay matrix, but its rows and columns are not the graph's vertices and arrows.
Not a graph of homomorphisms between irreducibles. Distinct complex irreducibles have no nonzero intertwiners between them. Edges record constituents after tensoring, not direct maps \(\rho_i\to\rho_j\).
Not automatically undirected or simple. Symmetry, absence of loops, and unit multiplicities require extra hypotheses; a general fixed representation may produce directed arrows, loops, and weights.
Not automatically connected. Faithfulness of \(V\) is the classical condition forcing every irreducible to appear in a tensor power and the graph to be connected.^[1]
Not the McKay correspondence itself. The graph is the representation-theoretic input. The classical correspondence identifies its special \(\mathrm{SU}(2)\) form with affine ADE and links it to quotient singularities and their resolutions.
Not a Dynkin diagram by definition. Most pairs \((G,V)\) do not produce an affine ADE diagram.
Not a mathematical flow graph. Both are weighted directed encodings, but a flow graph's nodes are variables and edges reproduce linear equations; a McKay graph's nodes are irreducibles and edges reproduce tensor decomposition.
Not a tensor product of graphs. The word “tensor” belongs to the representation operation being encoded, not to a graph-product construction.

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.^[3]
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.^[1]
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.^[2]
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.^[2]
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.

The classical node assumes complex representations of a finite group, where complete reducibility makes multiplicities canonical. In modular characteristic dividing \(|G|\), tensor products need not decompose into irreducibles as direct sums; composition-factor or indecomposable variants are related extensions, not automatic instances of the exact classical signature.

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.

The decomposition formula should appear before the picture. With arrows \(i\to j\) for constituents \(\rho_j\subset V\otimes\rho_i\), row \(i\) of \(A\) lists outgoing multiplicities. If a source instead puts those entries in column \(i\), transpose every matrix formula consistently. A label “2” and two parallel arrows carry the same multiplicity, but an unlabeled single edge does not.

Undirected drawings require an explicit reason. Frobenius reciprocity gives

\[ [V\otimes\rho_i:\rho_j]=[V^*\otimes\rho_j:\rho_i]. \]

Hence \(V\cong V^*\) makes \(A\) symmetric. It does not by itself remove loops or make all nonzero entries equal to one. Those additional properties hold in the nondegenerate finite-subgroup-of-\(\mathrm{SU}(2)\) defining-representation cases, not for every self-dual representation: \(G=\{\pm I\}\) has multiplicity-two adjacency, and the admitted trivial subgroup has two loops.^[3]

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.

A practical construction and audit loop is:

enumerate the irreducible characters and choose \(V\);
multiply \(\chi_V\chi_i\) and decompose it in the irreducible-character basis;
place the resulting coefficients in the declared row or column;
verify nonnegative integrality and the dimension equation \(Ad=(\dim V)d\);
if \(V\) is self-dual, verify symmetry rather than assuming it;
test connectivity only against the relevant faithfulness hypothesis; and
invoke ADE, Cartan, or singularity conclusions only after checking the \(G\subset\mathrm{SU}(2)\), defining-two-dimensional assumptions.

The main diagnostics are local and sharp. A dimension mismatch exposes a bad multiplicity row. A transpose mismatch reverses path semantics while leaving a plausible picture. A missing vertex means the irreducible roster is incomplete. An unexplained asymmetry signals either non-self-duality or an arithmetic error. A false ADE expectation is caught by checking the dimension, determinant-one, self-duality, loop, and multiplicity hypotheses before comparing diagrams.

Interventions follow the diagnosis: recompute a character inner product, restore a multiplicity label, transpose the entire convention consistently, add omitted irreducibles, replace \(G\) by the effective quotient when studying an unfaithful action, or state the result as a general McKay quiver rather than forcing a Dynkin identification.

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.

Several deductions become immediate. The \(k\)-step walk count is the coefficient of \([\rho_j]\) in \([V]^k[\rho_i]\). The positive dimension vector is an eigenvector because dimension is a ring homomorphism \(R(G)\to\mathbb Z\). Evaluation of characters at a fixed conjugacy class is another ring homomorphism to \(\mathbb C\), producing the character-value eigenvectors described by Steinberg.^[3] Duality transposes the operator: the matrix for \(V^*\) is \(A^T\).

For \(\dim V=2\), the matrix \(C=2I-A\) becomes the affine Cartan matrix only in the classical \(\mathrm{SU}(2)\) setting. There the dimension vector lies in its nullspace and the graph's combinatorics meets the ADE classification. The phrase “Cartan matrix of \(V\)” can be used more generally for \((\dim V)I-A\), but root-system conclusions do not follow from the notation alone.

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. A positive integer eigenvector with eigenvalue \(\dim V\) supplies a dimension sanity check. In the two-dimensional determinant-one habitat, an affine ADE shape signals the classical McKay correspondence and points toward the associated Kleinian singularity.

The generic graph skeleton transfers farther, but the node does not. A flow graph, social network, and dependency graph also use vertices and weighted edges, yet none makes an edge coefficient equal to an irreducible multiplicity in \(V\otimes\rho_i\). Outside representation theory, only Network and Representation travel; “McKay graph” does not.

Examples¶

Canonical: a directed McKay graph for \(C_3\)¶

Let \(G=C_3=\langle g\mid g^3=1\rangle\), and let \(\chi(g)=\omega\) for a primitive cube root \(\omega\). The irreducible complex representations are \(1,\chi,\chi^2\). Choose the fixed one-dimensional representation \(V=\chi\). Then

\[ \chi\otimes1=\chi,\qquad \chi\otimes\chi=\chi^2,\qquad \chi\otimes\chi^2=1. \]

With vertices ordered \((1,\chi,\chi^2)\),

\[ A=\begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&0 \end{pmatrix}. \]

The McKay graph is the directed 3-cycle \(1\to\chi\to\chi^2\to1\). Since \(A^3=I\), three tensorings return every irreducible to itself. The fixed representation is faithful, so the graph is connected, but \(V\not\cong V^*\), so the graph is not represented by paired opposite arrows.

Mapped back: the finite group is \(C_3\); the irreducible roster has three characters; the fixed tensoring representation is \(\chi\); the tensor-product decompositions are the three displayed equations; the direction convention reads left factor to resulting constituent; the multiplicity edges all have weight one; the McKay matrix is the permutation matrix; the walk interpretation gives \(A^3=I\); and the derived tests correctly predict connectedness without symmetry.

Applied / In Practice: the binary tetrahedral group and \(\widetilde E_6\)¶

Let \(G\) be the binary tetrahedral subgroup of \(\mathrm{SU}(2)\), of order 24, and choose its defining faithful two-dimensional representation \(V=2\). Its seven irreducibles have dimensions \(1,1,1,2,2,2,3\), whose squares sum to 24. Label the one-dimensional representations \(1,1',1''\), the corresponding two-dimensional representations \(2,2',2''\), and the three-dimensional representation \(3\). The tensor rules are

\[ \begin{aligned} V\otimes1&=2,&V\otimes1'&=2',&V\otimes1''&=2'',\\ V\otimes2&=1\oplus3,&V\otimes2'&=1'\oplus3,&V\otimes2''&=1''\oplus3,\\ V\otimes3&=2\oplus2'\oplus2''. \end{aligned} \]

These decompositions form a three-armed tree: central vertex \(3\), three adjacent dimension-2 vertices, and one outer dimension-1 vertex beyond each. This is the affine Dynkin diagram \(\widetilde E_6\). The dimension check is visible at every vertex: \(2\cdot3=2+2+2\), \(2\cdot2=3+1\), and \(2\cdot1=2\). The trivial representation is one of the three outer vertices and marks the extending node. This explicit case is tabulated in Stekolshchik's treatment of the McKay correspondence.^[4]

Mapped back: the finite group is binary tetrahedral; the irreducible roster is the seven displayed types; the fixed tensoring representation is the defining \(2\); the tensor-product decompositions are the seven equations; the direction convention is suppressed only after self-duality pairs all arrows; the multiplicity edges are unit edges; the McKay matrix is the \(\widetilde E_6\) adjacency matrix; the walk interpretation computes higher tensor powers; and the derived tests give the eigenvalue-2 dimension vector and the classical affine-ADE identification.^[1]^[3]

Structural Tensions¶

T1: Compact graph versus exact algebra. A drawing is easy to scan, but an omitted weight or reversed arrow changes a tensor decomposition. Diagnostic: reconstruct every \(V\otimes\rho_i\) from outgoing edges and compare it with the character calculation.

T2: Convention freedom versus matrix consistency. Global arrow reversal is harmless, while mixing row and column conventions invalidates path and spectrum formulas. Diagnostic: test a one-step decomposition and a two-step matrix power under the published convention before doing larger calculations.

T3: General quiver versus ADE special case. The famous undirected affine diagrams dominate intuition, but general McKay graphs may have loops, directions, weights, and components. Diagnostic: verify \(G\subset\mathrm{SU}(2)\) and the defining two-dimensional representation before importing ADE conclusions.

T4: Group identity versus representation choice. Saying “the graph of \(G\)” is convenient when a canonical action exists, but different \(V\) can produce radically different operators. Diagnostic: include \(V\) in the notation and metadata unless the context fixes it unambiguously.

T5: Self-duality versus simple undirectedness. Self-duality makes adjacency symmetric but does not, by itself, forbid loops or multiplicities. Diagnostic: check \(a_{ii}=0\) and \(a_{ij}\leq1\) independently from \(A=A^T\).

T6: Compression versus recoverability. One McKay matrix records multiplication by \([V]\), not the entire representation ring unless additional generators or spectral data suffice. Diagnostic: distinguish “tensor rules with this fixed \(V\)” from “all pairwise tensor-product rules.”

T7: Domain-specific autonomy versus reduction to parent structure. Representation, Network, and Decomposition explain encoding, graph carrier, and splitting into constituents, but they do not fix irreducible vertices, tensor multiplication, multiplicity adjacency, duality, or the McKay/ADE validity boundary. Diagnostic: retain this node when those representation-theoretic roles license deductions; reduce only a graph that lacks the exact tensor-decomposition semantics.

Structural–Framed Character¶

McKay Graph is structural-leaning domain-specific. Once \((G,V)\) is fixed, its adjacency is a formal invariant up to relabeling and global convention. Yet the node's recognition and use depend on specialist representation-theory semantics.

Evaluative weight: low; the graph records algebraic multiplicities rather than desirable or undesirable outcomes.
Human-practice-bound: no; tensor decompositions and character values do not depend on institutional use.
Institutional origin: no; the construction is mathematical rather than rule-governed by an organization.
Vocabulary travel: limited; irreducible representation, tensor product, constituent multiplicity, duality, character, and affine Dynkin diagram cannot be removed without changing the identity.
Import versus recognize: recognize; the graph exposes the multiplication-by-\([V]\) structure already present in \(R(G)\), although choosing \(V\) selects which operator to expose.

Its parent skeleton is a faithful graph representation of a decomposition operator. Its domain frame fixes the carrier basis as irreducible representations and the operator as tensoring by \(V\). Its character: formally structural in construction, irreducibly representation-theoretic in meaning.

Structural Core vs. Domain Accent¶

Skeletal structural core. A finite basis, a nonnegative linear operator, and a directed weighted graph are three views of the same data. Rows describe one-step transitions; powers describe iterated transitions; eigenvectors expose conserved or scaling quantities.

Indispensable domain accent. The basis elements are irreducible \(G\)-representations, the operator is multiplication by \([V]\) in the representation ring, coefficients are tensor-product multiplicities, dimension and character evaluation supply special eigenvectors, and self-duality and the \(\mathrm{SU}(2)\) hypothesis control the ADE boundary.

Why this is not a prime. Substitute states, web pages, or variables for irreducibles and the generic adjacency calculus survives, but the recognition test, decomposition calculation, duality test, character spectrum, and McKay correspondence vanish. The graph skeleton transfers; the named instrument and its licensed deductions do not.

McKay Graph instantiates Representation because it maps a tensoring operator into a graph medium with an exact readback. It instantiates Network because the node-edge topology, direction, loops, weights, reachability, and walks carry the calculation. It presupposes Decomposition because every adjacency row is obtained by splitting \(V\otimes\rho_i\) into irreducible constituents.

Matrix is a neighboring domain-specific surface: the McKay matrix and graph are equivalent encodings of the same operator, but Matrix is not an ancestor required in every graphical presentation. Isomorphism governs relabeling and correspondence claims but does not supply the tensor semantics. Mathematical Flow Graph is the strongest visual neighbor and the strongest wrong closure: both use weighted directed adjacency, yet their node and edge readbacks are incompatible.

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

Not to Be Confused With¶

Cayley graph. Its vertices are group elements and edges encode multiplication by group generators. Tell: Inspect a vertex label—an element belongs to a Cayley graph; an irreducible representation belongs to a McKay graph.
Character table. Its axes are irreducibles and conjugacy classes, and entries are character values. Tell: Ask whether entries are complex evaluations or nonnegative tensor multiplicities.
Fusion graph in a broader tensor category. This is the categorical generalization of the same multiplication-by-an-object idea, but its objects and semisimplicity conditions may not come from a finite group. Tell: Identify the underlying category and whether its vertices are \(\operatorname{Irr}(G)\).
McKay correspondence. The correspondence links representation theory, affine ADE data, and quotient-singularity geometry. Tell: A graph is one combinatorial artifact; a correspondence asserts a relationship between mathematical worlds.
McKay conjecture. This is a different theorem in finite-group character theory comparing counts of characters relative to a prime and a normalizer. Tell: Look for Sylow normalizers and character degrees rather than tensor adjacency.
Dynkin diagram. Only the special defining-representation graphs of finite \(\mathrm{SU}(2)\) subgroups yield affine ADE diagrams. Tell: Verify the subgroup and representation hypotheses before identifying the picture.
Mathematical Flow Graph. It encodes coupled equations with variable nodes and gain edges. Tell: Read one outgoing row: tensor constituents indicate McKay; coefficient contributions to variable equations indicate flow graph.
Graph of a representation homomorphism. The graph is not the set-theoretic graph of \(G\to\mathrm{GL}(V)\). Tell: McKay vertices enumerate all irreducible representation classes, not ordered input-output pairs of a function.
Tensor product of graphs. This combines graph vertex pairs under a graph-product rule. Tell: In a McKay graph, tensoring happens before graph construction inside \(V\otimes\rho_i\).

References¶

[1] John McKay, “Graphs, Singularities, and Finite Groups”, in The Santa Cruz Conference on Finite Groups, Proceedings of Symposia in Pure Mathematics 37, American Mathematical Society (1980), 183–186. Verified 2026-08-26. Primary source defining representation graphs by tensor-product multiplicities and establishing the classical finite-subgroup/graph observations, including faithfulness and connectedness. registry ↩a ↩b ↩c ↩d ↩e

[2] Lukas Bertsch, Ádám Gyenge, and Balázs Szendrői, “Kleinian Singularities: Some Geometry, Combinatorics and Representation Theory”, Jahresbericht der Deutschen Mathematiker-Vereinigung 126 (2024), 213–247. Verified 2026-08-26. Authoritative open survey giving the modern McKay-quiver definition, the self-dual paired-arrow argument, path-algebra context, and the precise affine-ADE/Kleinian-singularity boundary. registry ↩a ↩b ↩c ↩d

[3] Robert Steinberg, “Finite Subgroups of \(SU_2\), Dynkin Diagrams and Affine Coxeter Elements”, Pacific Journal of Mathematics 118.2 (1985), 587–598; full text. Verified 2026-08-26. Primary source for the McKay matrix, character-table eigenvectors, dimension vector, duality/symmetry criteria, and the affine ADE correspondence. registry ↩a ↩b ↩c ↩d ↩e ↩f

[4] Rafael Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence, Springer Monographs in Mathematics (2008), Appendix A. Verified 2026-08-26. Authoritative monograph tabulating the binary tetrahedral irreducibles and tensor decompositions that produce the \(\widetilde E_6\) example. registry ↩