Exceptional isomorphism

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In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms. [note 1] These coincidences are at times considered a matter of trivia, [1] but in other respects they can give rise to consequential phenomena, such as exceptional objects. [1] In the following, coincidences are organized according to the structures where they occur.

Contents

Groups

Finite simple groups

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are: [2]

Alternating groups and symmetric groups

The compound of five tetrahedra expresses the exceptional isomorphism between the chiral icosahedral group and the alternating group on five letters. Compound of five tetrahedra.png
The compound of five tetrahedra expresses the exceptional isomorphism between the chiral icosahedral group and the alternating group on five letters.

There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups: [3]

These can all be explained in a systematic way by using linear algebra (and the action of Sn on affine nspace) to define the isomorphism going from the right side to the left side. (The above isomorphisms for A8 and S8 are linked via the exceptional isomorphism SL4/μ2 ≅ SO6.)

There are also some coincidences with symmetries of regular polyhedra: the alternating group A5 agrees with the chiral icosahedral group (itself an exceptional object), and the double cover of the alternating group A5 is the binary icosahedral group.

Trivial group

The trivial group arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance:

Spheres

The spheres S0, S1, and S3 admit group structures, which can be described in many ways:

Spin groups

In addition to Spin(1), Spin(2) and Spin(3) above, there are isomorphisms for higher dimensional spin groups:

Also, Spin(8) has an exceptional order 3 triality automorphism.

Coxeter–Dynkin diagrams

There are some exceptional isomorphisms of Dynkin diagrams, yielding isomorphisms of the corresponding Coxeter groups and of polytopes realizing the symmetries, as well as isomorphisms of Lie algebras whose root systems are described by the same diagrams. These are:

Diagram Dynkin classification Lie algebra Polytope
CDel node.pngA1 = B1 = C1
CDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3x.pngCDel node.pngA2 = I2(2) 2-simplex is regular 3-gon (equilateral triangle)
CDel node.pngCDel 4.pngCDel node.pngBC2 = I2(4) 2-cube is 2-cross polytope is regular 4-gon (square)
CDel node.pngCDel node.pngCDel nodes.pngA1 × A1 = D2
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngA3 = D3 3-simplex is 3-demihypercube (regular tetrahedron)

See also

Notes

  1. Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions), but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).

References