7.6 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Exceptional object | 2/3 | https://en.wikipedia.org/wiki/Exceptional_object | reference | science, encyclopedia | 2026-05-05T07:23:52.521982+00:00 | kb-cron |
== 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
{\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).
== Outer automorphisms == 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
n
≥
3
,
n
≠
6
{\displaystyle n\geq 3,n\neq 6}
the alternating group
A
n
{\displaystyle A_{n}}
has one outer automorphism (corresponding to conjugation by an odd element of
S
n
{\displaystyle S_{n}}
) and the symmetric group
S
n
{\displaystyle S_{n}}
has no outer automorphisms. However, for
n
=
6
,
{\displaystyle n=6,}
there is an exceptional outer automorphism of
S
6
{\displaystyle S_{6}}
(of order 2), and correspondingly, the outer automorphism group of
A
6
{\displaystyle A_{6}}
is not
C
2
{\displaystyle C_{2}}
(the group of order 2), but rather
C
2
×
C
2
{\displaystyle C_{2}\times C_{2}}
, the Klein four-group. If one instead considers
A
6
{\displaystyle A_{6}}
as the (isomorphic) projective special linear group
PSL
(
2
,
9
)
{\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
A
6
≅
PSL
(
2
,
9
)
.
{\displaystyle A_{6}\cong \operatorname {PSL} (2,9).}
This exceptional outer automorphism is realized inside of the Mathieu group
M
12
{\displaystyle M_{12}}
and similarly,
M
12
{\displaystyle M_{12}}
acts on a set of 12 elements in 2 different ways.
Among Lie groups, the spin group
Spin
(
8
)
{\displaystyle \operatorname {Spin} (8)}
has an exceptionally large outer automorphism group (namely
S
3
{\displaystyle S_{3}}
), which corresponds to the exceptional symmetries of the Dynkin diagram
D
4
{\displaystyle D_{4}}
. This phenomenon is referred to as triality. The exceptional symmetry of the
D
4
{\displaystyle D_{4}}
diagram also gives rise to the Steinberg groups.
== Algebraic topology ==
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. 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. 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.
== Symmetric quantum measurements == 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. The sporadic SICs in dimension 8 are related to the integral octonions.
== Connections == 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.
=== 8 and 24 dimensions === Exceptional objects related to the number 8 include the following.
The octonions are 8-dimensional. The E8 lattice can be realized as the integral octonions (up to a scale factor). The exceptional Lie groups can be seen as symmetries of the octonions and structures derived from the octonions; further, the E8 algebra is related to the E8 lattice, as the notation implies (the lattice is generated by the root system of the algebra). Triality occurs for Spin(8), which also connects to 8 · 3 = 24. Likewise, exceptional objects related to the number 24 include the following.