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 ${\displaystyle S_{2}\wr S_{n}}$ 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 ${\displaystyle \{-n,-n+1,\cdots ,-1,1,2,\cdots ,n\}}$ or of the set ${\displaystyle \{-n,-n+1,\cdots ,n\}}$ such that ${\displaystyle \pi (i)=-\pi (-i)}$ 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 (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 (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.