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 B_n = C_n, 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 where 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, \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 n × n orthogonal matrices whose entries are all integers. Equivalently, this is the set of n × n 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 O × S_2 where O ≅ S_4 is the octahedral group, and S_2 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. 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 ), and one map coming from the parity of the permutation.

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