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. 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 S 2 ≀ S n {displaystyle S_{2}wr S_{n}} where S n {displaystyle S_{n}} 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, ..., −1, 1, 2, ..., n } or of the set { −n, −n + 1, ..., n } such that π(i) = −π(−i) 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 (Young 1930) according to (Kerber 1971, p. 2). In three dimensions, the hyperoctahedral group is known as O×S2 where O≅S4 is the octahedral group, and S2 is a symmetric group (equivalently, 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. 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 { ± 1 } {displaystyle {pm 1}} ), and one map coming from the parity of the permutation. Multiplying these together yields a third map C n → { ± 1 } {displaystyle C_{n} o {pm 1}} . The kernel of the first map is the Coxeter group D n . {displaystyle 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: