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 .
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 of the two-element group with the symmetric group , and may be represented as the set of invertible matrices with entries only , , or 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.
Low-dimensional examples The hyperoctahedral group in dimension is the group of symmetries of a line segment. This is a two-element group, consisting of the identity element and a single other element. In higher dimensions, hyperoctahedral groups have richer structure. Dimension 2: symmetries of a square
The hyperoctahedral group in two dimensions is the dihedral group of order , the symmetry group of a square. It has eight elements, of which four are rotations (including the identity, a rotation by an angle of ) and four are reflections. The group operation is functional composition; for example, first reflecting the square across the horizontal axis, then reflecting it across the diagonal of slope gives the same result as rotating the square by counter-clockwise, so in the group the product of these two reflections is that rotation.
The group can be characterized by generators and relations in several ways. One such expression is , where represents the identity transformation, represents the rotation by (of order ), represents any of the reflections (of order ), and the final relation captures the fact that reflecting across a line, then rotating by counter-clockwise, then reflecting across the same line has the same effect as applying a single rotation by clockwise.
A separate presentation of the group is . In this presentation, and represent two reflections, one across a diagonal and one across a line through the centers of two opposite sides. Then the product is a rotation by , of order .
Dimension 3: symmetries of a cube
Because the cube and regular octahedron are dual polyhedra, they have the same symmetry group. This group consists of elements: each symmetry is determined by picking a vertex of the cube, choosing the position of one of eight vertices to send it to, and choosing how its neighbors are assigned in one of ways.
Of the symmetries, several families are of note. Nine of the symmetries of the cube are reflections across a plane: three reflections across planes parallel to a pair of opposite faces, and six further reflections across the planes that pass through two opposite edges of the cube. Twenty-three of the symmetries are nontrivial rotations: nine are rotations around one of the lines that passes through the centers of two opposite faces (one each by , , and around the three axes), six are rotations by around a line through the midpoints of two opposite edges, and eight are rotations around one of the space diagonals (one each by and around the four diagonals). Along with the identity, these form the rotational subgroup of the cube. This subgroup is isomorphic to the symmetric group of permutations of a four-element set; for example, one can show that each permutation of the four space diagonals can be achieved by exactly one rotational symmetry. The remaining symmetries include the point reflection through the center of the cube that sends each vertex to the opposite vertex, and various improper rotations (a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis) of orders and .
If one chooses a Cartesian coordinate system so that the origin is at the center of the cube and the three coordinate axes are parallel to the edges, then the choice of where to send the single vertex can be achieved by reflections across the three coordinate planes; because these different reflections commute, these form a subgroup of the form , a direct product of three two-element groups. The rearrangements of the three neighbor vertices are given by a subgroup isomorphic to the symmetric group of permutations of a three-element set. Thus, the whole group of symmetries of the cube can be written as a semidirect product . The same group can also be written as the direct product of the rotational subgroup of the cube by the two-element group generated by the point reflection through the origin.
If one divides the cube into chambers by the planes fixed by each of its reflection symmetries, each chamber is bounded by three such planes. Calling the reflections across these planes , one can show that these form a generating set for all of the symmetries (that is, every other symmetry can be achieved by repeated compositions of these three symmetries), subject to the relations that , , , , , and are all the identity transformation.
In arbitrary dimension
For any positive integer , the -dimensional Euclidean space contains -dimensional analogues of the cube, called hypercubes. One such hypercube consists of all the points such that for , with vertices . The hyperoctahedral group consists of all rigid transformations of that send to itself: . Equivalently, one may consider the dual polytope of ; this is the -dimensional cross-polytope . It consists of all points in that satisfy the equation , and has as vertices the vectors , where is the standard basis vector of having th entry equal to and all other entries equal to .
The rigid transformations of that preserve the cross-polytope (equivalently, the hypercube ) are all linear transformations. In the standard basis for , the matrix of such a transformation must be a signed permutation matrix: it must have exactly one nonzero entry in each row and column, and the nonzero entries are all . Thus the hyperoctahedral group of dimension may be characterized as the group of signed permutation matrices under the operation of matrix multiplication. Combinatorially, the elements may be represented as signed permutations, that is, as -tuples that contain exactly one element from each of the sets , , ..., .
Algebraically, is isomorphic to the wreath product of the two-element group by the symmetric group . That is, it is a semidirect product of a direct product of copies of with the symmetric group: the normal subgroup acts by sign changes, while the symmetric group acts by permuting coordinates.
From these characterizations, one can see that the size of the hyperoctahedral group is , since there are (the factorial of ) ways to choose the positions of the nonzero entries (a permutation of the coordinate axes) and ways to choose whether each one should be positive or negative.
As a reflection, Coxeter, and Weyl group
A reflection in Euclidean space is a linear operator for which there is a hyperplane (that is, a subspace of dimension ) that fixes pointwise (that is, such that for all in ) and such that negates the vectors in the line perpendicular to . A finite group of invertible linear operators on is called a (finite real) reflection group if is generated by the reflections it contains. The hyperoctahedral group is a reflection group in this sense: the subgroup is generated by the sign changes, each of which fixes a coordinate hyperplane pointwise while negating the normal vector , and the subgroup is generated by its subset of transpositions, and the transposition fixes the hyperplane pointwise while negating its normal vector . Since is the product of these two subgroups, it is generated by the collection of both types of reflections.
For every finite real reflection group acting on a space of dimension , there is a standard procedure to produce a set of reflections that generate the group, subject to a simple collection of relations. The reflecting hyperplanes of the reflections divide space into a collection of regions called chambers. Each chamber has of the hyperplanes in its boundary, and the reflections across these planes form a generating set for the group.
In the case of the hyperoctahedral group, one such choice produces the reflections whose action on is given as follows: for any point in ,
and
for . That is, acts by negating the first coordinate, and acts by transposing the th and st coordinates for .
One can check that these reflections generate and that they satisfy the relations
if . In fact, one can show further that these are a complete set of relations; that is, that is the group with presentation
This gives the structure of a Coxeter group, with the corresponding set of simple reflections. The corresponding Coxeter–Dynkin diagram recording these relations is
with nodes.
As a Weyl group of a root system
The hyperoctahedral group arises as the symmetries of other geometric objects aside from polyhedra. A root system is a finite set of nonzero vectors (called roots) in Euclidean space that satisfy two properties: if belongs to then the scalar multiple belongs to only for , and if and belong to then so does the reflection of across the hyperplane orthogonal to . Each root system determines a finite real reflection group, generated by the reflections through the planes orthogonal to its roots.
A root system is crystallographic if its roots span a lattice. For there are, up to isomorphism, two crystallographic root systems whose associated Weyl group is : letting be the standard basis for , the type B root system consists of the vectors
whereas the type C root system is the following minor variant:
The root (or ) corresponds to the reflection that, when acting on a vector in , changes the sign of the th coordinate. The root corresponds to the reflection that swaps the th and th coordinates, while the root corresponds to the reflection that swaps the th and th coordinates and changes both of their signs.
In the case , these root systems are isomorphic, and both give the group . For general , the two root systems are not isomorphic, but they are dual to each other. For , the type B root system consists of the vectors from the center of a cube to the centers of its edges and faces, while the type C root system consists of the vectors from the center of an octahedron to its vertices and the centers of its edges.
As a permutation group
The hyperoctahedral group can be identified with the set of bijections from the set to itself that satisfy for all in , under the operation of functional composition. The bijection is determined by the signed permutation , which in this context is called the window notation of .
The representation of as a group of permutations of a set of size induces a natural inclusion map from the -dimensional hyperoctahedral group into the symmetric group on twice as many elements. The image of is the set of permutations in whose permutation matrix is fixed by rotation around its center. Equivalently, writing for the permutation in whose one-line notation is and whose cycle notation is , the image of is the set of permutations that satisfy , and also the set of permutations that commute with .
One may alternatively consider the group of bijections from the set to itself that satisfy for all , since the symmetry condition enforces . In this case the associated inclusion map is into the symmetric group .
Cycles and conjugacy classes