Binary Polyhedral Groups
The same classification applies to discrete subgroups of, the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin diagrams and the representations of these groups can be understood in terms of these diagrams. This connection is known as the McKay correspondence after John McKay. The connection to Platonic solids is described in (Dickson 1959). The correspondence uses the construction of McKay graph.
Note that the ADE correspondence is not the correspondence of Platonic solids to their reflection group of symmetries: for instance, in the ADE correspondence the tetrahedron, cube/octahedron, and dodecahedron/icosahedron correspond to while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter groups and
The orbifold of constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a du Val singularity.
The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a pair of binary polyhedral groups. This is known as the Slodowy correspondence – see (Stekolshchik 2008).
Read more about this topic: ADE Classification
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