Hyperoctahedral group


In mathematics, a hyperoctahedral group is an important type of group 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 n, the dimension of the hypercube.
As a Coxeter group it is of type Bn = Cn, and as a Weyl group it is associated to the orthogonal groups in odd dimensions. As a wreath product it is where is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set or of the set such that π = −π for all i. As a matrix group, it can be described as the group of n×n orthogonal matrices whose entries are all integers. The representation theory of the hyperoctahedral group was described by according to.
In three dimensions, the hyperoctahedral group is known as O×S2 where OS4 is the octahedral group, and S2 is a symmetric group of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex. In 4-dimensions it is called a hexadecachoric symmetry, after the regular 16-cell, or 4-orthoplex. In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.

By dimension

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 Dn and the symmetries of the demihypercube. 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", 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 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 below, and their intersection is the derived subgroup, of index 4, 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. In general, passing to the subquotient is the symmetry group of the projective demihypercube.
in three dimensions, order 24
The hyperoctahedral subgroup, Dn by dimension:
in three dimensions, order 24
in three dimensions, order 24
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, stabilizes at the Klein four-group, and is given by:
This is easily seen directly: the elements are order 2, and all conjugate, as are the transpositions in , 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", and the sign of the permutation. Multiplying these together yields a third non-trivial map, and together with the trivial map these form the 4-group.

H2: Schur multipliers

The second homology groups, known classically as the Schur multipliers, were computed in.
They are: