Exceptional object

The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) — making five Platonic solids. In four dimensions, a total of six regular polytopes exist, including the 120-cell, the 600-cell and the 24-cell. There are no other regular polytopes, as the only regular polytopes in higher dimensions are of the hypercube, simplex, orthoplex series. In all dimensions combined, there are therefore three series and five exceptional polytopes.[5]

Moreover, the pattern is similar if non-convex polytopes are included: in two dimensions, there is a regular star polygon for every rational number ${\displaystyle \textstyle p/q>2}$.[6] In three dimensions, there are four Kepler–Poinsot polyhedra, and in four dimensions, ten Schläfli–Hess polychora; in higher dimensions, there are no non-convex regular figures.

These can be generalized to tessellations of other spaces, especially uniform tessellations, notably tilings of Euclidean space (honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above, the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross-polytope, and hypercube lattice.

Schwarz triangles

Further information: Schwarz triangle

(3 3 2)	(4 3 2)	(5 3 2)
(3 3 3)	(4 4 2)	(6 3 2)

Related to tilings and the regular polyhedra, there are exceptional Schwarz triangles (triangles that tile the sphere, or more generally Euclidean plane or hyperbolic plane via their triangle group of reflections in their edges), particularly the Möbius triangles. In the sphere, there are 3 Möbius triangles (and 1 1-parameter family), corresponding to the 3 exceptional Platonic solid groups, while in the Euclidean plane, there are 3 Möbius triangles, corresponding to the 3 special triangles: 60-60-60 (equilateral), 45-45-90 (isosceles right), and 30-60-90. There are additional exceptional Schwarz triangles in the sphere and Euclidean plane. By contrast, in the hyperbolic plane, there is a 3-parameter family of Möbius triangles, and none exceptional.

The relations between the sporadic groups, most being related to the monster.

The finite simple groups have been classified into a number of series as well as 26 sporadic groups.[7] Of these, 20 are subgroups or subquotients of the monster group, referred to as the "Happy Family", while 6 are not, and are referred to as "pariahs".

Several of the sporadic groups are related to the Leech lattice, most notably the Conway group Co₁, which is the automorphism group of the Leech lattice, quotiented out by its center.

There are only three finite-dimensional 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.

The simple Lie groups form a number of series (classical Lie groups) labelled A, B, C and D. In addition, there are 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 thus the exceptional Lie groups correspond to exceptional root systems and exceptional Dynkin diagrams.

There are a few exceptional objects with supersymmetry. The classification of superalgebras by Kac and Tierry-Mieg indicates that the Lie superalgebras G(3) in 31 dimensions and F(4) in 40 dimensions, and the Jordan superalgebras K₃ and K₁₀, are examples of exceptional objects.[8][9]

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 without 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.

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 ${\displaystyle M_{24}}$, 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 ${\displaystyle M_{24}}$.

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

The codewords of the extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors. So every 24-bit word with weight 5 can be corrected to a codeword with weight 8. The bits of a 24-bit word can be thought of as specifying the possible subsets of a 24 element set. So the extended binary Golay code gives a unique 8 element subset for each 5 element subset. In fact, it defines S(5,8,24).

Certain families of groups often have a certain outer automorphism group, but in particular cases, they have other exceptional outer automorphisms.

Among families of finite simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for ${\displaystyle n\geq 3,n\neq 6}$ the alternating group ${\displaystyle A_{n}}$ has one outer automorphism (corresponding to conjugation by an odd element of ${\displaystyle S_{n}}$) and the symmetric group ${\displaystyle S_{n}}$ has no outer automorphisms. However, for ${\displaystyle n=6,}$ there is an exceptional outer automorphism of ${\displaystyle S_{6}}$ (of order 2), and correspondingly, the outer automorphism group of ${\displaystyle A_{6}}$ is not ${\displaystyle C_{2}}$ (the group of order 2), but rather ${\displaystyle C_{2}\times C_{2}}$, the Klein four-group.[10][11][12]

If one instead considers ${\displaystyle A_{6}}$ as the (isomorphic) projective special linear group ${\displaystyle \operatorname {PSL} (2,9)}$, then the outer automorphism is not exceptional; thus the exceptional-ness can be seen as due to the exceptional isomorphism ${\displaystyle A_{6}\cong \operatorname {PSL} (2,9).}$ This exceptional outer automorphism is realized inside of the Mathieu group ${\displaystyle M_{12}}$ and similarly, ${\displaystyle M_{12}}$ acts on a set of 12 elements in 2 different ways.

Among Lie groups, the spin group ${\displaystyle \operatorname {Spin} (8)}$ has an exceptionally large outer automorphism group (namely ${\displaystyle S_{3}}$), which corresponds to the exceptional symmetries of the Dynkin diagram ${\displaystyle D_{4}}$. This phenomenon is referred to as triality.

The exceptional symmetry of the ${\displaystyle D_{4}}$ diagram also gives rise to the Steinberg groups.

The Kervaire invariant is an invariant of a (4k + 2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. More specifically, the Kervaire invariant applies to a framed manifold, that is, to a manifold equipped with an embedding into Euclidean space and a trivialization of the normal bundle. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open.[13][14] These five or six framed cobordism classes of manifolds having Kervaire invariant 1 are exceptional objects related to exotic spheres. The first three cases are related to the complex numbers, quaternions and octonions respectively: a manifold of Kervaire invariant 1 can be constructed as the product of two spheres, with its exotic framing determined by the normed division algebra.[15]

Due to similarities of dimensions, it is conjectured that the remaining cases (dimensions 30, 62 and 126) are related to the Rosenfeld projective planes, which are defined over algebras constructed from the octonions. Specifically, it has been conjectured that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower, but this remains unconfirmed.[16]

In quantum information theory, there exist structures known as SIC-POVMs or SICs, which correspond to maximal sets of complex equiangular lines. Some of the known SICs—those in vector spaces of 2 and 3 dimensions, as well as certain solutions in 8 dimensions—are considered exceptional objects and called "sporadic SICs". They differ from the other known SICs in ways that involve their symmetry groups, the Galois theory of the numerical values of their vector components, and so forth.[17] The sporadic SICs in dimension 8 are related to the integral octonions.[18]

Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to 8 and 24 dimensions, noting that 24 = 8 · 3. By contrast, the pariah groups stand apart, as the name suggests.

• Exceptional isomorphism