In
mathematics, a hyperoctahedral group is an important type of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
that can be realized as the
group of symmetries of a
hypercube or of a
cross-polytope. It was named by
Alfred Young in 1930. Groups of this type are identified by a parameter , the dimension of the hypercube.
As a
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
it is of type , and as a
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
it is associated to the
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
s and with the
orthogonal groups in odd dimensions. As a
wreath product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...
it is
where is the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
of degree . As a
permutation group, the group is the signed symmetric group of permutations ''π'' either of the set or of the set such that for all . As a
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
, it can be described as the group of
orthogonal matrices whose entries are all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. Equivalently, this is the set of matrices with entries only 0, 1, or –1, which are
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral group was described by according to .
In three dimensions, the hyperoctahedral group is known as where is the
octahedral group
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
, and is a symmetric group (here a
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
) of order 2. Geometric figures in three dimensions with this symmetry group are said to have
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
, named after the regular
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, or 3-
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
. In 4-dimensions it is called a
hexadecachoric symmetry, after the regular
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...
, or 4-
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
. In two dimensions, the hyperoctahedral group structure is the abstract
dihedral group of order eight, describing the symmetry of a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, or 2-orthoplex.
By dimension
![3-ary Boolean functions; cube permutations; 5](https://upload.wikimedia.org/wikipedia/commons/1/15/3-ary_Boolean_functions%3B_cube_permutations%3B_5.svg)
Hyperoctahedral groups can be named as
Bn, a bracket notation, or as a Coxeter group graph:
Subgroups
There is a notable index two subgroup, corresponding to the Coxeter group ''D''
''n'' and the symmetries of the
demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''- polytopes constructed from alternation of an ''n''- hypercube, labeled as ''hγn'' for being ''half'' of the hy ...
. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the ''n'' copies of
), and one map coming from the parity of the permutation. Multiplying these together yields a third map
. The kernel of the first map is the Coxeter group
In terms of
signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.
The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in
''H''1: Abelianization below, and their intersection is the
derived subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest norma ...
, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.
In the other direction, the center is the subgroup of scalar matrices, ; geometrically, quotienting out by this corresponds to passing to the
projective orthogonal group.
In dimension 2 these groups completely describe the hyperoctahedral group, which is the
dihedral group Dih4 of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2). In general, passing to the subquotient (derived subgroup, mod center) is the symmetry group of the projective demihypercube.
![Sphere symmetry group td](https://upload.wikimedia.org/wikipedia/commons/a/a5/Sphere_symmetry_group_td.png)
The hyperoctahedral subgroup, D
n by dimension:
![Sphere symmetry group o](https://upload.wikimedia.org/wikipedia/commons/2/2d/Sphere_symmetry_group_o.png)
The chiral hyper-octahedral symmetry, is the direct subgroup, index 2 of hyper-octahedral symmetry.
Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:
These groups have ''n'' orthogonal mirrors in ''n''-dimensions.
Homology
The
group homology of the hyperoctahedral group is similar to that of the symmetric group, and exhibits stabilization, in the sense of
stable homotopy theory.
H1: abelianization
The first homology group, which agrees with the
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
, stabilizes at the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one. ...
, and is given by:
:
This is easily seen directly: the
elements are order 2 (which is non-empty for
), and all conjugate, as are the transpositions in
(which is non-empty for
), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to
as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the ''n'' copies of
), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix, which sends both these classes to
), and together with the trivial map these form the 4-group.
H2: Schur multipliers
The second homology groups, known classically as the
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
s, were computed in .
They are:
:
Notes
References
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{{refend
Finite reflection groups