Hyperoctahedral group

Groups of this type are identified by a parameter n, the dimension of the hypercube.

The representation theory of the hyperoctahedral group was described by (Young 1930) according to (Kerber 1971, p. 2).

Hyperoctahedral groups can be named as Bn, a bracket notation, or as a Coxeter group graph: 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

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 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:[5] These groups have n orthogonal mirrors in n-dimensions.

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

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

The maps are explicitly given as "the product of the signs of all the elements" (in the n copies of

Multiplying these together yields a third non-trivial map (the determinant of the matrix, which sends both these classes to

The second homology groups, known classically as the Schur multipliers, were computed in (Ihara & Yokonuma 1965).