Signed symmetric group

A hyperoctahedral group is a type of mathematical group that arises 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 it is of type , 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 is the 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, it can be described as the group of orthogonal matrices whose entries are all integers. Equivalently, this is the set of 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 where is the octahedral group, and 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 in the $n$-th dimension are isomorphic to $S_2 \wr S_n$ ($\wr$ denots the Wreath product) and can be named as B_n, 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. 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 $C_n \to \$. 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 ''H''₁: 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, ; 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, D_n 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:
:$H_1(C_n, \mathbf) = \begin 0 & n = 0\\ \mathbf/2 & n = 1\\ \mathbf/2 \times \mathbf/2 & n \geq 2 \end.$
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 $-1 \in \,$ 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 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:
:$H_2(C_n,\mathbf) = \begin 0 & n = 0, 1\\ \mathbf/2 & n = 2\\ (\mathbf/2)^2 & n = 3\\ (\mathbf/2)^3 & n \geq 4 \end.$

Notes

References

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Finite reflection groups