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 $C2$, and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is $$S_2 \wr S_n$$ where $Sn$ 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 $\{-n, -n+1, \cdots, -1, 1, 2, \cdots, n\}$ or of the set $\{-n, -n+1, \cdots, n\}$ such that $\pi(i) = -\pi(-i)$ for all $i$. As a matrix group, it can be described as the group of $C3$ orthogonal matrices whose entries are all integers. Equivalently, this is the set of $Oh$ matrices with entries only 0, 1, or –1, which are invertible, 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 $Bn = Cn$ where $n × n$ is the octahedral group, and $n × n$ is a symmetric group (here a cyclic 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" (in the n copies of $$\{\pm 1\}$$), and one map coming from the parity of the permutation. Multiplying these together yields a third map $$C_n \to \{\pm 1\}$$. The kernel of the first map is the Coxeter group $$D_n.$$ 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 H1: Abelianization below, and their intersection is the derived subgroup, 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, {±1}; 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. The hyperoctahedral subgroup, Dn by dimension:





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:
 * $$H_1(C_n, \mathbf{Z}) = \begin{cases} 0 & n = 0\\

\mathbf{Z}/2 & n = 1\\ \mathbf{Z}/2 \times \mathbf{Z}/2 & n \geq 2 \end{cases}.$$ This is easily seen directly: the $$-1$$ elements are order 2 (which is non-empty for $$n\geq 1$$), and all conjugate, as are the transpositions in $$S_n$$ (which is non-empty for $$n\geq 2$$), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to $$-1 \in \{\pm 1\},$$ as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the n copies of $$\{\pm 1\}$$), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the determinant of the matrix, which sends both these classes to $$-1$$), 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:
 * $$H_2(C_n,\mathbf{Z}) = \begin{cases}

0 & n = 0, 1\\ \mathbf{Z}/2 & n = 2\\ (\mathbf{Z}/2)^2 & n = 3\\ (\mathbf{Z}/2)^3 & n \geq 4 \end{cases}.$$