Exceptional object

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects as well as a finite number of exceptions that don't fit into any series. These are known as exceptional objects.

Frequently these exceptional objects play a further and important role in the subject. Surprisingly, the exceptional objects in one branch of mathematics are often related to the exceptional objects in others.

Regular polyhedra

The prototypical examples of exceptional objects arise when we classify the regular polyhedra. In 2 dimensions we have a series of n-gons for n≥3. In every dimension above 2 we find analogues of the cube, tetrahedron and octahedron. In 3 dimensions we find two more polyhedra making 5 platonic solids. In 4 dimensions we have a total of 6 regular polyhedra including the 120-cell, the 600-cell and 24-cell. There are no other regular polyhedra. So we have two series and 5 exceptional polyhedra.

Finite simple groups

The finite simple groups have been classified into a number of series as well as 26 sporadic groups.

Division algebras

There are only three associative division algebras over the reals - the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the algebra of octonions. The octonions are connected to a wide variety of exceptional objects. For example the exceptional formally real Jordan algebra is the Albert algebra of 3 by 3 self-adjoint matrices over the octonions.

Simple Lie groups

The simple lie groups form a number of series (classical Lie groups) labelled A, B, C and D. In addition we have the exceptional groups G₂ (the automorphism group of the octonions), F₄, E₆, E₇, E₈. These last four groups can be viewed as the symmetry groups of projective planes over O, C⊗O, H⊗O and O⊗O respectively, where O is the octonions and the tensor products are over the reals.

The classification of Lie groups corresponds to the classification of root systems and so the exceptional Lie groups correspond to exceptional root systems and exceptional Dynkin diagrams.

Unimodular lattices

Up to isometry there is only one even unimodular lattice in 15 dimensions or less — the E₈ lattice. Up to dimension 24 there is only one even unimodular lattice with no roots, the Leech lattice. Three of the sporadic simple groups were discovered by Conway while investigating the automorphism group of the Leech lattice. For example Co₁ is the automorphism group itself modulo ±1. The groups Co₂ and Co₃, as well as a number of other sporadic groups, arise as stabilisers of various subsets of the Leech lattice.

Codes

Some codes also stand out as exceptional objects, in particular the perfect binary Golay code which is closely related to the Leech lattice. The Mathieu group $M\_\{24\}$, one of the sporadic simple groups, is the group of automorphisms of the extended binary Golay code and four more of the sporadic simple groups arise as various types of stabilizer subgroup of $M\_\{24\}$.

Block designs

An exceptional block design is the Steiner system S(5,8,24) whose automorphism group is the sporadic simple Mathieu group $M\_\{24\}$.

References

* [http://www.findarticles.com/p/articles/mi_qa3742/is_199811/ai_n8816298 Exceptional Objects] , John Stilwell, American Mathematical Monthly, Nov 1998
* [http://math.ucr.edu/home/baez/week106.html This Week's Finds in Mathematical Physics, Week 106] , John Baez
* [http://math.ucr.edu/home/baez/platonic.html Platonic Solids in all Dimensions] , John Baez