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, 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 reflectionss. 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, 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.

. 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 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, 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 of the rotational subgroup of the cube by the two-element group generated by the point reflection through the origin.

are all the identity transformation.

In arbitrary dimension

For any positive integer , the -dimensional Euclidean space contains -dimensional analogues of the cube, called hypercubes. The hyperoctahedral group consists of all rigid transformations . Equivalently, one may consider the dual polytope; this is the -dimensional cross-polytope is the standard basis vector having th entry equal to and all other entries equal to .

) are all linear transformations. In the standard basis, 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 . 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 , , ..., .

is isomorphic to the wreath product by the symmetric group. That is, it is a semidirect product of a direct product of with the symmetric group: the normal subgroup acts by permuting coordinates.

(the factorial of ways to choose whether each one should be positive or negative.

As a reflection, Coxeter, and Weyl group

A reflection is a linear operator for which there is a hyperplane (that is, a subspace of dimension ) that for all in ) and such that negates the vectors in the line perpendicular to . A finite group of invertible is called a (finite real) reflection group if is generated by the reflections it contains. is generated by the pointwise while negating the normal vector is generated by its subset of transpositions pointwise while negating its normal vector 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.

acts by transposing the th and .

is the group with presentation the structure of a Coxeter group 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 , 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., the type B root system whereas the type C root system, changes the sign of the corresponds to the reflection that swaps the th and corresponds to the reflection that swaps the th and th coordinates and changes both of their signs.

In the case . 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

can be identified with the set of bijections for all , under the operation of functional composition. The bijection , which in this context is called the window notation of .

as a group of permutations of a set of size from the -dimensional hyperoctahedral group into the symmetric group on twice as many elements. The image of whose permutation matrix is fixed by rotation around its center., the image of .

for all .

Cycles and conjugacy classes

When viewed as a signed permutation in window notation, a cycle of an element is a cycle in the underlying permutation that we get by erasing all minus signs; the length of the cycle is the number of entries it contains. A cycle is positive if the number of negative numbers among its entries in the window is even, and negative otherwise., each positive cycle corresponds to two cycles, one containing the negatives of the entries of the other, while each negative cycle corresponds to a single cycle that contains if and only if it contains . has three cycles: the negative cycles and and the positive cycle , its cycle decomposition is .

The cycle type of a signed permutation of two integer partitions where consists of the lengths of the positive cycles of and consists of the lengths of the negative cycles of . Two signed permutations and belong to the same conjugacy class (that is, there exists another signed permutation ) if and only if they have the same cycle type. In other words, the conjugacy classes where and are two integer partitions whose parts sum to .

are conjugate if and only if they have the same number of cycles of each length, and their images under the inclusion also have the same number of cycles of each length.

Special elements

In total, the hyperoctahedral group of dimension reflections. Of these, are the sign-change reflections reflections are the transposition-like (that is, it is the result of transposing the entries and .

The length of an element of a Coxeter group with respect to a set of simple reflections is the smallest number such that of elements of . The longest element, this element is . That is, it is the point reflection through the origin, whose matrix is the negative of the identity matrix. Combinatorially, its window notation is . (a special case of the fact that the length of the longest element is always equal to the number of reflections in a finite real reflection group).

In a finite real reflection group , a Coxeter element is the product of the simple reflections in any simple system . The Coxeter elements all belong to a single conjugacy class.; that is, the elements with a single negative -cycle. If is the longest element.

An element of a reflection group is said to be a quasi-Coxeter element of of the factors is a generating set for , the quasi-Coxeter elements are precisely the Coxeter elements and their conjugates.

A signed permutation to a permutation if and only if every cycle of is even. is the double factorial.

Subgroups

The hyperoctahedral group has several notable families of subgroups.

Center

The center consists solely of the longest element and the identity.

Index-2 subgroups

has three subgroups of index ): the orientation-preserving symmetries of the hypercube, the even-signed permutation group (the Coxeter group of type D), and the generalized alternating group.

The alternating subgroup or even subgroup of the hyperoctahedral group is the subgroup consisting of elements of determinant in the matrix representation, that is, the elements that can be written as a product of an even number of reflections. They are also the orientation-preserving symmetries of the hypercube.

A second index-. It is also the symmetry group of the demihypercube.

When by its center.

The third index- of the two-element group with the alternating group of even permutations (a generalized alternating group).

Commutator subgroup

, the commutator subgroup has index ; it is equal to the commutator subgroup of the even-signed subgroup of type D.

Parabolic subgroups

The standard parabolic subgroups are the Young subgroups, etc. among themselves....

Associated Lie groups and algebras

In the theory of Lie groups (or the related theory of algebraic groups), every connected, compact Lie group has an associated finite group, called its Weyl group, that plays an important role in its representation theory. is the Weyl group of the "type B" and "type C" objects in the classification: the special orthogonal group and the symplectic group . Likewise, in the theory of Lie algebras, every complex simple Lie algebra.

Inversions, descents, and length

of adjacent transpositions, the length of a permutation . For a signed permutation of where has the same meaning. ("negative sum pairs"). The generating function.

A simple reflection is a (right) descent of an element . For a signed permutation of length , the set of descents of . the number of descents of a signed permutation by number of descents (its Eulerian polynomial is the Eulerian polynomial for the symmetric group. The exponential generating function was given by .

Polynomial invariants

For every subgroup , its action extends to an action on the polynomial ring of variables linearly and then substituting. The ring of invariants of the hyperoctahedral group has a simple description: it consists precisely of the symmetric polynomials of the variables.

for some algebraically independent set of homogeneous polynomials. These basic invariants, these degrees are , , ..., . One choice of basic invariants is the elementary symmetric polynomials. Another choice is given by the even power sum symmetric polynomials.

Representation theory

The representation theory of the hyperoctahedral group was worked out by Alfred Young, in the paper , as an application of his study of the representation theory of symmetric groups. This was subsequently generalized by Wilhelm Specht to the representation theory of more general wreath products.

Concretely, the isomorphism classes of irreducible representations of the sizes is equal to . The irreducible character is the induced representation is the tensor product is the linear character that takes the value and takes the value is a partition of is a partition of . These characterizations can be combined with the Murnaghan–Nakayama rule to give a combinatorial formula for character values. And there is a characteristic map from the character ring to a ring of symmetric functions in two sets of variables.

The projective representations likewise have a close relationship to the projective and linear representations of the symmetric group.

Bruhat order and weak order

The Bruhat order is a natural order on every Coxeter group, defined as the transitive closure of the relation that whenever there is a reflection . (The name Bruhat order arises from the connection with the Bruhat decomposition if and only if one can get from to by applying a sequence of transpositions, each of which swaps two entries such that the smaller entry is to the left in the one-line notation. where .

There is another simple criterion to tell whether one permutation is less than another in Bruhat order, in terms of comparing the number of s in certain regions of the permutation matrices of and . The analogous criterion in the hyperoctahedral group is as follows: given a signed permutation of size , where is taken to be . Then for signed permutations and of size , where is the inclusion map defined in .

For any Coxeter group with simple generating set , there are two subposets of the Bruhat order called the left and right weak orders. These posets are not the same, but they are isomorphic: an element is less than in the left weak order, so any property of one can be deduced from the corresponding property of the other. may be characterized in several equivalent ways:

• when there is a simple reflection in ;

• ; and

• if and only if the (right) inversion set is contained in the right inversion set of ,

among others.

In the case of a permutation coincides with the notion of inversions described above, namely, a transposition is a (right) inversion of , its right inversion set may be decomposed into three pieces, that correspond to the sets

• ,

• , and

• ,

.

Absolute order and noncrossing partitions

In a reflection group , the reflection length of an element is the smallest number in such that of the reflections. In a finite real reflection group acting on a space of dimension , the reflection length of an element is equal to is the fixed space of . is viewed as a signed permutation, this fixed space dimension is equal to the number of positive cycles of . That is, if the conjugacy class of is is the number of parts of the partition .

In a reflection group , the absolute order on and the -noncrossing partitions is the interval in absolute order that lies below a Coxeter element of (as in ).-noncrossing partitions and the set partitions that are noncrossing, in the sense that when the numbers are drawn in order on the boundary of a circle and the parts of the partition are drawn as their convex hulls, no two parts intersect. Moreover, under this correspondence, the order relation corresponds to the refinement order on set partitions., the lattice of noncrossing partitions is isomorphic to the sublattice of noncrossing partitions of a -element set that are symmetric under rotation.

The number of noncrossing partitions of an -element set is the th Catalan number, with the number of partitions at each rank given by a Narayana number. is the central binomial coefficient, with the number of elements at rank .

Tableau insertion algorithms

is associated to a Young diagram boxes in the second row, and so on. A standard Young tableau of shape is a filling of the Young diagram of with the numbers , so that numbers increase in each row from left to right and in each column from smaller-numbered rows to larger-numbered rows. The Robinson–Schensted correspondence and the set of pairs of standard Young tableaux such that the shapes of and are the same partition of . It provides a crucial connection between combinatorics, the theory of symmetric functions, and the representation theory of the symmetric group.

There are several versions of the RobinsonSchensted correspondence for the hyperoctahedral group. In one of these, the role of standard Young tableaux is played by bitableaux: if and are two partitions of total size , a standard bitableau of shape are fillings of the Young diagrams of and , respectively, and the two tableaux together are filled with are bitableaux of the same shape (a bipartition of ). The bijection can be realized essentially by applying the usual (symmetric group) RobinsonSchensted correspondence separately to the positive and negative entries of the window notation of a signed permutation.

Associated polyhedra

In addition to its connection with the hypercube and the cross-polytope, the hyperoctahedral group is associated with other polyhedra. The -dimensional polytope whose vertices are all the permutations of is the type-B permutohedron, a signed analogue of the permutohedron. (A combinatorially equivalent polytope can be constructed by taking the convex hull.) Its 1-skeleton is the Hasse diagram. it is an octagon it is the truncated cuboctahedron.

The cyclohedron is the type-B analogue of the associahedron. Just as the facets of the associahedron can be indexed by sets of noncrossing diagonals of an -gon, the facets of the cyclohedron can be indexed by centrally symmetric sets of noncrossing diagonals of a -gon. Its -skeleton is a contraction of the -skeleton of the type-B permutohedron.

Homology

The group homology of the hyperoctahedral group exhibits stabilization, in the sense of stable homotopy theory, and so in particular they are eventually independent of . (This is true regardless of the choice of coefficients.)

have been explicitly computed for small is the same as the abelianization. For any Coxeter group , the abelianization is a elementary 2–group, with stabilization at the Klein four-group.

(also known as the Schur multipliers.

Other properties

is a subgroup of the orthogonal group orthogonal matrices. It consists exactly of those orthogonal matrices whose entries are all integers.

has a minimal generating set of size .

For any finite group , the strong symmetric genus of is the smallest genus of an orientable surface on which acts faithfully (when the genus is ( instead of ( instead of ).

Generalizations and extensions

Colored permutations

of -colored permutations monomial matrices whose nonzero entries are complex th roots of unity, these groups are no longer real reflection groups; instead, they belong to the infinite family of imprimitive complex reflection groups. is generated by complex reflections: the is a primitive th root of unity.

Affine Coxeter groups

Each finite Weyl group is associated to an affine Coxeter group, which is generated by the group together with a reflection across a translated copy of one of the reflecting hyperplanes of a reflection in of order , there is (up to isomorphism) a unique associated affine Coxeter group. The lines of the reflections in the group divide the plane into an infinite number of congruent isosceles right triangles (the tetrakis square tiling). is the highest root of the root system and is the standard dot product.

have CoxeterDynkin diagrams

1. and ,

respectively.

that commute with the reflection across and also the reflection across (and consequently with the infinite dihedral group generated by these two reflections). By construction, all permutations in this combinatorial model fix the multiples of . In either model, such a permutation is completely determined by its .

, the affine group of type B is the subgroup of the group of type C consisting of those permutations for which the number of positive integers such that is negative is even. (This number is always finite.) Both affine groups can furthermore be realized as subgroups of an appropriate affine symmetric group.

Braid group

Each Coxeter group is associated to an Artin–Tits group for each generator . In particular, the Artin is generated by (and no others).

ArtinTits groups are sometimes also known as generalized braid groups, because the Artin of the (finite) symmetric group is the braid group on strands. Not all ArtinTits groups have a natural representation in terms of geometric braids. However, the Artin strands consisting of those braids for which a particular strand ends in the same position it started in, or equivalently as the braid group of strands in an annular region. The Artin is also isomorphic to the mapping class group of a closed disc with marked points, with a single point fixed pointwise, with an infinite cyclic group.

Infinite hyperoctahedral group

, fixing all integers of magnitude larger than . Then the infinite hyperoctahedral group of these groups (or in other words as the direct limit of the sequence of inclusions). for all .

Tables and diagrams

The Coxeter–Dynkin diagram is

with nodes; the corresponding Coxeter bracket notation is

1. .

The CoxeterDynkin diagrams of the associated affine Coxeter groups are

and

.

The following table collects various "numerological" data attached to the hyperoctahedral groups.

See also

• Hexadecachoric symmetry, the four-dimensional case (symmetry group of the tesseract and 16-cell)

• 5-cube, 5-orthoplex, 6-cube, 6-orthoplex, the five- and six-dimensional regular polytopes with hyperoctahedral symmetry

Notes

Works cited