Table of Contents
Hyperoctahedral group
A hyperoctahedral group is any member of a sequence 
of classical orthogonal matrix groups.
Definition
As a matrix group
For every 
the hyperoctahedral group 
for dimension 
can be defined as the subgroup of the general linear group 
given by all orthogonal 
-matrices with integer entries, i.e., the set
where, if 
, then 
is the complex conjugate of 
and 
the transpose and where 
is the identity 
-matrix.
As a wreath product
The hyperoctahedral group for dimension , where 
, can also be defined as the wreath product of groups 
with respect to the natural action 
, where 
is the symmetric group and where 
is the cyclic group of order 
.
Given a set 
, two groups 
and 
and a group action 
of on the (unrestricted) wreath product 
of and with respect to 
is the semi-direct product of groups of the direct product group 
over 
and with respect to the group homomorphism 
.
If 
and 
are groups and 
a group homomorphism from 
to the group of group automorphisms of 
, then the (outer) semi-direct product 
of and with respect to 
is the group with underlying set 
, the cartesian product of and , and with group law 
.
For the hyperoctahedral group this means that 
is given by the group with underlying set 
and group law 
.
Basic properties
The hyperoctahedral groups 
are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. More precisely, it is the category of partitions with blocks of even size that induces the corepresentation categories of . Its canonical generating set of partitions is 
.
Representation theory
Cohomology
Related quantum groups
References
