Hyperoctahedral group

The hyperoctahedral groups are a family of mathematical groups that arise as the group of symmetries of the square, the cube, and their higher-dimensional counterparts (the hypercubes), as well as the corresponding dual polytopes (the regular octahedron and its higher-dimensional counterparts, the cross-polytopes). There is one hyperoctahedral group for each dimension n.

In addition to their role in geometry, the hyperoctahedral groups also appear in Lie theory, as the Weyl groups associated to the symplectic groups and the orthogonal groups and their associated Lie algebras, and in combinatorics, where they may be viewed as a signed version of the symmetric groups, with their elements given by signed permutations. Algebraically, each hyperoctahedral group may be realized as a wreath product${\displaystyle C_{2}\wr S_{n}}$ of the two-element group ${\displaystyle C_{2}}$ with the symmetric group ${\displaystyle S_{n}}$, and may be represented as the set of invertible matrices with entries only 0, 1, or −1 and with exactly one non-zero entry in each row or column. The family of hyperoctahedral groups forms type B in the classification of finite Coxeter groups.

The hyperoctahedral groups were named by Alfred Young in 1930.