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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 where 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 or of the set 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 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 (equivalently, cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry. In two dimensions, the hyperoctahedral group is known as the dihedral group of order eight, describing the symmetry of a square.
== By dimension ==
Hyperoctahedral groups can be named as BCn, a bracket notation, or as a Coxeter group graph:
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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