groups.qft.org
Search⌘ K

Find the exact object.

Names, aliases, Unicode, and common TeX notation. Open full search

Finite groups & geometry

From a finite group to an ADE diagram

The quaternion group connects SU(2) tensor products to affine D₄.

Fix the defining representation

For a finite subgroup Γ of SU(2), restrict the defining two-dimensional SU(2) representation to Γ and call it V. Tensoring V with each irreducible representation Ri defines multiplicities aij through V⊗Ri = ⊕j aij Rj.

V⊗Ri≅⨁jaijRjV\otimes R_i\cong\bigoplus_j a_{ij}R_j

Read the representation graph

For Q₈ there are four one-dimensional irreps and one two-dimensional irrep. Tensoring the doublet with any singlet returns the doublet; tensoring the doublet with itself yields all four singlets. The resulting graph is affine D₄.

2⊗2=1⊕1a⊕1b⊕1c,2⊗1i=2\mathbf2\otimes\mathbf2=\mathbf1\oplus\mathbf1_a\oplus\mathbf1_b\oplus\mathbf1_c,\qquad\mathbf2\otimes\mathbf1_i=\mathbf2

Connect algebra and geometry

The binary tetrahedral, octahedral, and icosahedral subgroups give affine E₆, E₇, and E₈ graphs. Removing the affine node leads to the finite diagrams associated with the exceptional curves in resolutions of the corresponding surface singularities.

Keep the action with the group

The chosen inclusion into SU(2) supplies the defining doublet. The site treats that action as explicit data so a representation graph can be traced back to the tensor products that produce it.

The affine D4 McKay graph of Q8The two-dimensional representation is connected to each of four one-dimensional representations.211a1b1c
Tensoring with the defining doublet gives the affine D₄ graph. The chosen finite subgroup of SU(2) and its defining action determine this construction.

Sources and further reading

This guide connects sourced reference material with supported calculations. Source curation and automated checks do not constitute completed independent expert review. Read the editorial method.