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 orthogonal groups in odd dimensions. As a wreath product it is ${\displaystyle S_{2}\wr S_{n}}$ where ${\displaystyle S_{n}}$ 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).

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