Hyperoctahedral group

The nine symmetry planes of a cube

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 Template:Mvar.

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 $C_2\wr S_n$ of the two-element group $C_2$ with the symmetric group $S_n$, and may be represented as the set of invertible matrices with entries only Template:Math, Template:Math, or Template:Math 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.Template:Sfnp

Low-dimensional examples

The hyperoctahedral group in dimension Template:Math 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

Template:Main The hyperoctahedral group in two dimensions is the dihedral group of order Template:Math, the symmetry group of a square. It has eight elements, of which four are rotations (including the identity, a rotation by an angle of Template:Math) 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 Template:Math gives the same result as rotating the square by Template:Math counter-clockwise, so in the group the product of these two reflections is that rotation.Template:Sfnp Template:Multiple image The group can be characterized by generators and relations in several ways. One such expression is $⟨r,s\mid r^4=s^2=1,srs=r^{-1}⟩$, where Template:Math represents the identity transformation, Template:Mvar represents the rotation by Template:Math (of order Template:Math), Template:Mvar represents any of the reflections (of order Template:Math), and the final relation captures the fact that reflecting across a line, then rotating by Template:Math counter-clockwise, then reflecting across the same line has the same effect as applying a single rotation by Template:Math clockwise.Template:SfnpTemplate:Sfnp

A separate presentation of the group is $⟨s,t\mid s^2=t^2=(st{)}^4=1⟩$. In this presentation, Template:Mvar and Template:Mvar represent two reflections, one across a diagonal and one across a line through the centers of two opposite sides. Then the product Template:Math is a rotation by Template:Math, of order Template:Math.Template:SfnpTemplate:Sfnp

Dimension 3: symmetries of a cube

The cube and the regular octahedron are dual polyhedra, with the same group of symmetries; illustration from Harmonice Mundi by Johannes Kepler (1619).

Because the cube and regular octahedron are dual polyhedra, they have the same symmetry group.Template:Sfnp This group consists of Template:Math 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 Template:Math ways.Template:Sfnp

Of the Template:Math 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.Template:SfnpTemplate:Sfnp 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 Template:Math, Template:Math, and Template:Math around the three axes), six are rotations by Template:Math around a line through the midpoints of two opposite edges, and eight are rotations around one of the space diagonals (one each by Template:Math and Template:Math around the four diagonals). Along with the identity, these form the rotational subgroup of the cube. This subgroup is isomorphic to the symmetric group $S_4$ 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.Template:Sfnp 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 Template:Math and Template:Math.Template:SfnpTemplate:Sfnp

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 $C_2\times C_2\times C_2$, 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 $S_3$ of permutations of a three-element set. Thus, the whole group of symmetries of the cube can be written as a semidirect product $(C_2\times C_2\times C_2)⋊S_3$. The same group can also be written as the direct product $C_2\times S_4$ of the rotational subgroup of the cube by the two-element group generated by the point reflection through the origin.Template:Sfnp

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 $s_0,s_1,s_2$, 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 $s_0^2$, $s_1^2$, $s_2^2$, $(s_0{s}_2{)}^2$, $(s_1{s}_2{)}^3$, and $(s_0{s}_1{)}^4$ are all the identity transformation.Template:Sfnp

In arbitrary dimension

For any positive integer Template:Mvar, the Template:Mvar-dimensional Euclidean space ${ℝ}^n$ contains Template:Mvar-dimensional analogues of the cube, called hypercubes. One such hypercube $\mathscr{H}$ consists of all the points $(x_1,\dots ,x_n)$ such that $|x_i|\le 1$ for $i=1,\dots ,n$, with vertices $(\pm 1,\dots ,\pm 1)$.Template:SfnpTemplate:Sfnp The hyperoctahedral group consists of all rigid transformations Template:Mvar of ${ℝ}^n$ that send $\mathscr{H}$ to itself: $w(\mathscr{H})=\mathscr{H}$. Equivalently, one may consider the dual polytope of $\mathscr{H}$; this is the Template:Mvar-dimensional cross-polytope $𝒪$. It consists of all points in ${ℝ}^n$ that satisfy the equation $|x_1|+\dots +|x_n|\le 1$, and has as vertices the vectors $\pm e_1,\dots ,\pm e_n$, where $e_i=(0,\dots ,0,1,0,\dots ,0)$ is the standard basis vector of ${ℝ}^n$ having Template:Mvarth entry equal to Template:Math and all other entries equal to Template:Math.Template:Sfnp

The rigid transformations of ${ℝ}^n$ that preserve the cross-polytope $𝒪$ (equivalently, the hypercube $\mathscr{H}$) are all linear transformations. In the standard basis for ${ℝ}^n$, 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 Template:Math.Template:Efn Thus the hyperoctahedral group $S_n^{\pm }$ of dimension Template:Mvar may be characterized as the group of Template:Math signed permutation matrices under the operation of matrix multiplication.Template:Sfnp Combinatorially, the elements may be represented as signed permutations, that is, as Template:Mvar-tuples $[w(1),w(2),\dots ,w(n)]$ that contain exactly one element from each of the Template:Mvar sets Template:Math, Template:Math, ..., Template:Math.Template:Sfnp

Algebraically, $S_n^{\pm }$ is isomorphic to the wreath product $C_2\wr S_n$ of the two-element group $C_2=\{+1,-1\}$ by the symmetric group $S_n$. That is, it is a semidirect product $C_2^n⋊S_n$ of a direct product $C_2^n=C_2\times \cdots \times C_2$ of Template:Mvar copies of $C_2$ with the symmetric group: the normal subgroup $C_2^n$ acts by sign changes, while the symmetric group $S_n$ acts by permuting coordinates.Template:Sfnp

From these characterizations, one can see that the size of the hyperoctahedral group is $2^{n}n!$, since there are $n!$ (the factorial of Template:Mvar) ways to choose the positions of the nonzero entries (a permutation of the coordinate axes) and $2^n$ ways to choose whether each one should be positive or negative.Template:Sfnp

As a reflection, Coxeter, and Weyl group

A reflection in Euclidean space ${ℝ}^n$ is a linear operator $t:{ℝ}^n\to {ℝ}^n$ for which there is a hyperplane Template:Mvar (that is, a subspace of dimension Template:Math) that Template:Mvar fixes pointwise (that is, such that $t(x)=x$ for all Template:Mvar in Template:Mvar) and such that Template:Mvar negates the vectors in the line perpendicular to Template:Mvar. A finite group Template:Mvar of invertible linear operators on ${ℝ}^n$ is called a (finite real) reflection group if Template:Mvar is generated by the reflections it contains.Template:Sfnp The hyperoctahedral group is a reflection group in this sense: the subgroup $C_2^n$ is generated by the Template:Mvar sign changes, each of which fixes a coordinate hyperplane $\{(x_1,\dots ,x_n):x_i=0\}$ pointwise while negating the normal vector $e_i$, and the subgroup $S_n$ is generated by its subset of transpositions, and the transposition $(ij)$ fixes the hyperplane $\{(x_1,\dots ,x_n):x_i=x_j\}$ pointwise while negating its normal vector $e_i-e_j$. Since $S_n^{\pm }$ is the product of these two subgroups, it is generated by the collection of both types of reflections.Template:SfnpTemplate:Sfnp

For every finite real reflection group acting on a space of dimension Template:Mvar, there is a standard procedure to produce a set of Template:Mvar 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.Template:Sfnp Each chamber has Template:Mvar of the hyperplanes in its boundary, and the reflections across these planes form a generating set for the group.Template:Sfnp

In the case of the hyperoctahedral group, one such choice produces the reflections $s_0,s_1,\dots ,s_{n-1}$ whose action on ${ℝ}^n$ is given as follows: for any point $x=(x_1,\dots ,x_n)$ in ${ℝ}^n$, $$s_0(x)=(-x_1,x_2,\dots ,x_n)$$ and $$s_i(x)=(x_1,x_2,\dots ,x_{i-1},x_{i+1},x_i,x_{i+2},x_{i+3},\dots ,x_n)$$ for $i=1,\dots ,n-1$. That is, $s_0$ acts by negating the first coordinate, and $s_i$ acts by transposing the Template:Mvarth and Template:Mathst coordinates for $i=1,\dots ,n-1$.Template:SfnpTemplate:Efn

One can check that these reflections generate $S_n^{\pm }$ and that they satisfy the relations $$(s_0{s}_1{)}^4=1,(s_i{s}_{i+1}{)}^3=1,\text{and }(s_i{s}_j{)}^2=1$$ if $|i-j|>1$. In fact, one can show further that these are a complete set of relations; that is, that $S_n^{\pm }$ is the group with presentation $$⟨s_0,s_1,\dots ,s_{n-1}\mid (s_i{)}^2=1,(s_0{s}_1{)}^4=1,(s_i{s}_{i+1}{)}^3=1\text{ for }i=1,\dots ,n-1,\text{ and }(s_i{s}_j{)}^2=1\text{ if }|i-j|>1⟩.$$ This gives $S_n^{\pm }$ the structure of a Coxeter group, with $S=\{s_0,\dots ,s_{n-1}\}$ the corresponding set of simple reflections.Template:Sfnp The corresponding Coxeter–Dynkin diagram recording these relations is

Template:Math

with Template:Mvar nodes.Template:SfnpTemplate:Sfnp

As a Weyl group of a root system

The root systems of types B₃ and C₃. Only the tip of each vector is drawn; the origin is at the center of the cube/octahedron.

The hyperoctahedral group arises as the symmetries of other geometric objects aside from polyhedra. A root system Template:Math is a finite set of nonzero vectors (called roots) in Euclidean space that satisfy two properties: if Template:Mvar belongs to Template:Math then the scalar multiple Template:Mvar belongs to Template:Math only for $c=\pm 1$, and if Template:Mvar and Template:Mvar belong to Template:Math then so does the reflection of Template:Mvar across the hyperplane orthogonal to Template:Mvar.Template:Sfnp Each root system determines a finite real reflection group, generated by the reflections through the planes orthogonal to its roots.Template:Sfnp

A root system is crystallographic if its roots span a lattice.Template:EfnTemplate:SfnpTemplate:Sfnp For $n\ge 3$ there are, up to isomorphism, two crystallographic root systems whose associated Weyl group is $S_n^{\pm }$: letting $e_1,\dots ,e_n$ be the standard basis for ${ℝ}^n$, the type B root system ${\mathrm{\Phi }}_{\mathrm{B}}$ consists of the vectors $${\mathrm{\Phi }}_{\mathrm{B}}=\{e_1,\dots ,e_n\}\bigcup \{\pm e_i\pm e_j:1\le i<j\le n\}$$ whereas the type C root system ${\mathrm{\Phi }}_{\mathrm{C}}$ is the following minor variant: $${\mathrm{\Phi }}_{\mathrm{C}}=\{2e_1,\dots ,2e_n\}\bigcup \{\pm e_i\pm e_j:1\le i<j\le n\}.$$ The root $e_i$ (or $2e_i$) corresponds to the reflection that, when acting on a vector $(x_1,\dots ,x_n)$ in ${ℝ}^n$, changes the sign of the Template:Mvarth coordinate. The root $e_i-e_j$ corresponds to the reflection that swaps the Template:Mvarth and Template:Mvarth coordinates, while the root $e_i+e_j$ corresponds to the reflection that swaps the Template:Mvarth and Template:Mvarth coordinates and changes both of their signs.Template:Sfnp

In the case Template:Math, these root systems are isomorphic, and both give the group $S_2^{\pm }\cong D_{2\times 4}$. For general Template:Mvar, the two root systems are not isomorphic, but they are dual to each other.Template:EfnTemplate:SfnpTemplate:Sfnp For Template:Math, 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.Template:Sfnp

As a permutation group

The hyperoctahedral group $S_n^{\pm }$ can be identified with the set of bijections Template:Mvar from the set $\{-n,-n+1,\dots ,-1,1,2,\dots ,n\}$ to itself that satisfy $w(-i)=-w(i)$ for all Template:Mvar in $\{-n,-n+1,\dots ,-1,1,2,\dots ,n\}$, under the operation of functional composition. The bijection Template:Mvar is determined by the signed permutation $[w(1),w(2),\dots ,w(n)]$, which in this context is called the window notation of Template:Mvar.Template:SfnpTemplate:Efn

The representation of $S_n^{\pm }$ as a group of permutations of a set of size Template:Math induces a natural inclusion map $\iota :S_n^{\pm }\to S_{2n}$ from the Template:Mvar-dimensional hyperoctahedral group into the symmetric group on twice as many elements. The image of Template:Mvar is the set of permutations in $S_{2n}$ whose permutation matrix is fixed by Template:Math rotation around its center.Template:Sfnp Equivalently, writing $w_0$ for the permutation in $S_{2n}$ whose one-line notation is $(2n,2n-1,\dots ,2,1)$ and whose cycle notation is $(12n)(22n-1)\cdots (nn+1)$, the image of Template:Mvar is the set of permutations $g$ that satisfy $w_{0}g{w}_0=g$, and also the set of permutations that commute with $w_0$.Template:SfnpTemplate:Sfnp

One may alternatively consider the group of bijections from the set $[-n,n]\cap \mathbb{Z}=\{-n,-n+1,\dots ,n-1,n\}$ to itself that satisfy $w(-i)=-w(i)$ for all Template:Mvar, since the symmetry condition enforces $w(0)=0$. In this case the associated inclusion map is into the symmetric group $S_{2n+1}$.Template:Sfnp

Cycles and conjugacy classes

When viewed as a signed permutation in window notation, a cycle of an element Template:Mvar of $S_n^{\pm }$ 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.Template:SfnpTemplate:Efn When viewed as a permutation of $\{-n,\dots ,-1,1,\dots ,n\}$, 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 Template:Mvar if and only if it contains Template:Math.Template:Sfnp For example, the signed permutation with window notation $[-3,5,1,-7,2,8,-6,-4]$ has three cycles: the negative cycles Template:Math and Template:Math and the positive cycle Template:Math. As a permutation of $\{-8,\dots ,-1,1,\dots ,8\}$, its cycle decomposition is Template:Math.Template:SfnpTemplate:Sfnp

The cycle type of a signed permutation Template:Mvar is the pair $(\lambda ,\mu )$ of two integer partitions where Template:Mvar consists of the lengths of the positive cycles of Template:Mvar and Template:Mvar consists of the lengths of the negative cycles of Template:Mvar. Two signed permutations Template:Mvar and Template:Mvar belong to the same conjugacy class of $S_n^{\pm }$ (that is, there exists another signed permutation Template:Mvar such that $gu{g}^{-1}=w$) if and only if they have the same cycle type. In other words, the conjugacy classes in $S_n^{\pm }$ are indexed by pairs $(\lambda ,\mu )$ where Template:Mvar and Template:Mvar are two integer partitions whose parts sum to Template:Mvar.Template:Sfnp

Equivalently, two signed permutations in $S_n^{\pm }$ are conjugate if and only if they have the same number of cycles of each length, and their images under the inclusion Template:Mvar into the symmetric group $S_{2n}$ also have the same number of cycles of each length.Template:Sfnp

Special elements

In total, the hyperoctahedral group of dimension Template:Mvar has $n^2$ reflections. Of these, Template:Mvar are the sign-change reflections across a coordinate hyperplane $\{(x_1,\dots ,x_n):x_i=0\}$. They are given in cycle notation by $\{(i-i):1\le i\le n\}$ and in window notation by $\{[1,\dots ,i-1,-i,i+1,\dots ,n]:i=1,\dots ,n\};$ they form a single conjugacy class, indexed by the pair of integer partitions $(⟨1^{n-1}⟩,⟨1⟩)$. The other $n^2-n$ reflections are the transposition-like reflections, across the hyperplanes $\{(x_1,\dots ,x_n):x_i=x_j\}$ or $\{(x_1,\dots ,x_n):x_i=-x_j\}$ for $1\le i<j\le n$. They are given in cycle notation by $$\{(ij)(-i-j):1\le i<|j|\le n\}.$$ In window notation, the reflection across the plane with equation $x_i=x_j$ is given by $[1,2,\dots ,i-1,j,i+1,\dots ,j-1,i,j+1,j+2,\dots ,n]$ (that is, it is a transposition) while the reflection across the plane with equation $x_i+x_j=0$ is given by $[1,2,\dots ,i-1,-j,i+1,\dots ,j-1,-i,j+1,j+2,\dots ,n]$ (that is, it is the result of transposing the entries Template:Mvar and Template:Mvar and also changing both of their signs). Together, they form the conjugacy class indexed by $(⟨2,1^{n-2}⟩,\varnothing )$.Template:SfnpTemplate:Sfnp

The length ${\ell }_S(g)$ of an element Template:Mvar of a Coxeter group Template:Mvar with respect to a set Template:Mvar of simple reflections is the smallest number Template:Mvar such that Template:Mvar can be written as a product $g=s_{i_1}\cdots s_{i_k}$ of Template:Mvar elements of Template:Mvar.Template:Sfnp The longest element in a finite real reflection group is the (always unique) element whose Coxeter length is as large as possible. In $S_n^{\pm }$, this element is Template:Math. That is, it is the point reflection through the origin, whose matrix is the negative of the identity matrix.Template:Sfnp Combinatorially, its window notation is Template:Math.Template:Sfnp Its Coxeter length is $n^2$ (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).Template:Sfnp

In a finite real reflection group Template:Mvar, a Coxeter element is the product of the simple reflections in any simple system Template:Math. The Coxeter elements all belong to a single conjugacy class.Template:SfnpTemplate:Sfnp In $S_n^{\pm }$, this is the conjugacy class indexed by $(\lambda ,\mu )=(\varnothing ,⟨n⟩)$; that is, the elements with a single negative Template:Mvar-cycle.Template:Sfnp If Template:Mvar is a Coxeter element in $S_n^{\pm }$ then $c^n=-1$ is the longest element.Template:EfnTemplate:Sfnp

An element Template:Mvar of a reflection group Template:Mvar is said to be a quasi-Coxeter element if there is a factorization $c=t_1\cdot t_2\cdots t_k$ of Template:Mvar as a product of the minimum number $k={\ell }_R(c)$ of reflections such that the set $\{t_1,t_2,\dots ,t_k\}$ of the factors is a generating set for Template:Mvar. In $S_n^{\pm }$, the quasi-Coxeter elements are precisely the Coxeter elements and their conjugates.Template:Sfnp

A signed permutation Template:Mvar is conjugate in $S_n^{\pm }$ to a permutation if and only if every cycle of Template:Mvar is even.Template:Sfnp The number of such elements in $S_n^{\pm }$ is the double factorial $(2n-1)!!=(2n-1)\cdot (2n-3)\cdots 3\cdot 1$.Template:Sfnp

Subgroups

The hyperoctahedral group has several notable families of subgroups.

Center

The center of $S_n^{\pm }$ consists solely of the longest element Template:Math and the identity.Template:Sfnp

Index-Template:Math subgroups

Template:See also For $n\ge 2$, the group $S_n^{\pm }$ has three subgroups of index Template:Math (that is, subgroups that include exactly half the elements in $S_n^{\pm }$):Template:Sfnp 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 Template:Math 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.Template:Sfnp

A second index-Template:Math subgroup is formed by those elements whose signed permutation has an even number of minus signs. This group is again a Coxeter group, of type D. One possible set of generators for this subgroup is $s_0{s}_1{s}_0=(1-2)(-12),s_1,s_2,\dots ,s_{n-1}$, where the $s_i$ are the Coxeter generating set for $S_n^{\pm }$.Template:Sfnp It is also the symmetry group of the demihypercube, the polytope that one gets by taking the convex hull of every other vertex of the hypercube $\mathscr{H}$.Template:Sfnp

When Template:Mvar is odd, the hyperoctahedral group $S_n^{\pm }$ is the direct product of its even subgroup and its center, and also the product of the type-D subgroup with its center. Moreover, in this case the even subgroup and type-D subgroup are isomorphic and are further isomorphic to the quotient of $S_n^{\pm }$ by its center.Template:Sfnp

The third index-Template:Math subgroup is the wreath product $C_2\wr A_n$ of the two-element group with the alternating group of even permutations (a generalized alternating group).Template:Sfnp

Commutator subgroup

For $n\ge 2$, the commutator subgroup of $S_n^{\pm }$ has index Template:Math; it is equal to the commutator subgroup of the even-signed subgroup of type D.Template:Sfnp

Parabolic subgroups

The standard parabolic subgroups of $S_n$ are the Young subgroups: for each tuple $(a_1,\dots ,a_k)$ of positive integers such that $a_1+\dots +a_k=n$, the Young subgroup associated to the tuple is the subgroup isomorphic to $S_{a_1}\times \cdots \times S_{a_k}$ that arises by permuting the values in each of the sets $\{1,2,\dots ,a_1\}$, $\{a_1+1,a_1+2,\dots ,a_1+a_2\}$, etc. among themselves.Template:Sfnp The standard parabolic subgroups of $S_n^{\pm }$ are the subgroups of the form $S_{a_0}^{\pm }\times S_{a_1}\times \cdots S_{a_k}$ where $a_0$ is a nonnegative integer and $a_1,\dots ,a_k$ are positive integers such that $a_0+a_1+\dots +a_k=n$.Template:Sfnp In particular, the maximal standard parabolic subgroups of the hyperoctahedral group $S_n^{\pm }$ are the setwise stabilizers of the set $\{k,\dots ,n\}$ for $k=1,\dots ,n$.Template:Sfnp That is, each maximal standard parabolic is isomorphic to the direct product $S_{k-1}^{\pm }\times S_{n+1-k}$ of a smaller hyperoctahedral group and a symmetric group. When $k=1$, we have as a special case that the symmetric group $S_n$ is a maximal parabolic subgroup of $S_n^{\pm }$. Meanwhile, the hyperoctahedral group $S_k^{\pm }$ is a parabolic subgroup of $S_n^{\pm }$ for all $k\le n$.Template:Sfnp

Associated Lie groups and algebras

Template:See also 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.Template:Sfnp The hyperoctahedral group $S_n^{\pm }$ is the Weyl group of the "type B" and "type C" objects in the classification: the special orthogonal group Template:Math and the symplectic group Template:Math.Template:Sfnp Likewise, in the theory of Lie algebras, every complex simple Lie algebra has an associated Weyl group; $S_n^{\pm }$ is the Weyl group of the Lie algebras $𝔰𝔬(2n+1)$ and $𝔰𝔭(n)$.Template:Sfnp

Inversions, descents, and length

In the symmetric group $S_n$, with respect to the generating set $S=\{(12),(23),\dots ,(n-1n)\}$ of adjacent transpositions, the length of a permutation Template:Mvar is given by $${\ell }_S(w)=\mathrm{inv}(w(1)\cdots w(n))$$ where $\mathrm{inv}(w(1)\cdots w(n))$ is the number of pairs $i,j\in \{1,\dots ,n\}$ such that $i<j$ and $w(i)>w(j)$.Template:Sfnp For a signed permutation Template:Mvar in $S_n^{\pm }$, its length with respect to the Coxeter generating set $S=\{(1-1),(12)(-1-2),\dots ,(n-1n)(1-n-n)\}$ of Template:Slink can be computed as $${\ell }_S(w)=\mathrm{inv}(w(1)\cdots w(n))-\sum _{\{j:1\le j\le n,w(j)<0\}}w(j),$$ where Template:Math has the same meaning.Template:Sfnp This may also be written as $${\ell }_S(w)=\frac{\mathrm{inv}(w(-n)\cdots w(-1)w(1)\cdots w(n))+\mathrm{neg}(w(1)\cdots w(n))}{2},$$ where $\mathrm{neg}(w(1)\cdots w(n))$ is the number of negative values among $\{w(1),\dots ,w(n)\}$, and as $${\ell }_S(w)=\mathrm{inv}(w(1)\cdots w(n))+\mathrm{neg}(w(1)\cdots w(n))+\mathrm{nsp}(w(1)\cdots w(n)),$$ where $\mathrm{nsp}(w(1)\cdots w(n))$ is the number of pairs $\{i,j\}\subset \{1,\dots ,n\}$ such that $w(i)+w(j)<0$ ("negative sum pairs").Template:Sfnp The generating function for $S_n^{\pm }$ by length is $$\sum _{w\in S_n^{\pm }}{q}^{{\ell }_S(w)}=(1+q)(1+q+q^2+q^3)\cdots (1+q+\dots +q^{2n-1})$$ for any $n\ge 1$.Template:Sfnp

A simple reflection Template:Mvar is a (right) descent of an element Template:Mvar in a Coxeter group if $\ell (ws)=\ell (w)-1$.Template:Sfnp For a signed permutation Template:Mvar of length Template:Mvar, the set of descents of Template:Mvar is $$\{s_i:w(i)>w(i+1)\},$$ where we take by convention $w(0)=0$.Template:Sfnp Denoting by $\mathrm{des}(w)$ the number of descents of a signed permutation Template:Mvar, the generating function $$B_n(t)=\sum _{w\in S_n^{\pm }}{t}^{\mathrm{des}(w)}$$ for $S_n^{\pm }$ by number of descents (its Eulerian polynomial) satisfies the following identities: $$B_n(t)=(1+(2n-1)t)B_{n-1}(t)+2t(1-t)B{\text{'}}_{n-1}(t),$$ $$\frac{B_n(t)}{(1-t{)}^{n+1}}=\sum _{k\ge 0}(2k+1{)}^n{t}^k,$$ $$B_n(t)=(1+t)B_{n-1}(t)+2t\sum _{i=1}^{n-1}{2}^i(\genfrac{}{}{0ex}{}{n-1}{i})B_{n-1-i}(t)\cdot A_i(t),$$ and $$2B_n(t^2)=(1+t{)}^{n+1}{A}_n(t)+(1-t{)}^{n+1}{A}_n(-t),$$ where $$A_n(t)=\sum _{w\in S_n}{t}^{\mathrm{des}(w)}$$ is the Eulerian polynomial for the symmetric group.Template:Sfnp The exponential generating function for the $B_n(t)$ is $$\sum _{n\ge 0}{B}_n(t)\frac{z^n}{n!}=\frac{(t-1)e^{z(t-1)}}{t-e^{2z(t-1)}}.$$ A trivariate generating function for the descent number and length of signed permutations over $S_n^{\pm }$ for all $n\ge 0$ was given by Template:Harvtxt.Template:Sfnp

Polynomial invariants

For every subgroup Template:Mvar of the orthogonal group on ${ℝ}^n$, its action extends to an action on the polynomial ring $R=ℝ[x_1,\dots ,x_n]$ by acting on the vector $(x_1,\dots ,x_n)$ of variables linearly and then substituting. The ring of invariants of such a group is the subring $R^G=\{f\in R:w(f)=f\text{ for all }w\in G\}$. The ring of invariants $R^{S_n^{\pm }}$ of the hyperoctahedral group has a simple description: it consists precisely of the symmetric polynomials in the squares $x_1^2,\dots ,x_n^2$ of the variables.Template:Sfnp

Since $S_n^{\pm }$ is a reflection group, the ring of invariants is itself a polynomial ring; that is, $R^{S_n^{\pm }}=ℝ[p_1,p_2,\dots ,p_n]$ for some algebraically independent set of homogeneous polynomials $p_1,\dots ,p_n$. These basic invariants are not uniquely determined by the group, but their degrees are. For the hyperoctahedral group $S_n^{\pm }$, these degrees are Template:Math, Template:Math, ..., Template:Math. One choice of basic invariants is the elementary symmetric polynomials in the squares of the variables: $$\sum _{1\le i_1<\dots <i_k\le n}{x}_{i_1}^2\cdots x_{i_k}^2$$ for $k=1,\dots ,n$.Template:Sfnp Another choice is given by the even power sum symmetric polynomials $$x_1^{2k}+x_2^{2k}+\dots +x_n^{2k}$$ for $k=1,\dots ,n$.Template:Sfnp

Representation theory

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

Concretely, the isomorphism classes of irreducible representations of $S_n^{\pm }$ are indexed by pairs $(\lambda ,\mu )$ of integer partitions for which the sum $|\lambda |+|\mu |$ of the sizes is equal to Template:Mvar. The irreducible character ${\chi }^{\lambda ,\varnothing }$ indexed by $(\lambda ,\varnothing )$ is the induced representation of the symmetric group character indexed by $\lambda$ from the subgroup $S_n$ to $S_n^{\pm }$; the character indexed by $(\varnothing ,\mu )$ is the tensor product $\delta \otimes {\chi }^{\mu ,\varnothing }$ where $\delta$ is the linear character that takes the value Template:Math on transpositions in $S_n^{\pm }$ and takes the value Template:Math on sign-change reflections; and if $\lambda$ is a partition of Template:Mvar and $\mu$ is a partition of Template:Math, then ${\chi }^{\lambda ,\mu }$ is the induction product of ${\chi }^{\lambda ,\varnothing }\otimes {\chi }^{\varnothing ,\mu }$ from $S_k^{\pm }\times S_{n-k}^{\pm }$ to $S_n^{\pm }$. 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.Template:Sfnp

The projective representations of $S_n^{\pm }$ likewise have a close relationship to the projective and linear representations of the symmetric group.Template:Sfnp

Bruhat order and weak order

The Hasse diagram of the Bruhat order for the hyperoctahedral group Template:Math. The edges that correspond to multiplication on the right by a simple reflection are thickened.

The Bruhat order is a natural order on every Coxeter group, defined as the transitive closure of the relation that Template:Math whenever there is a reflection Template:Mvar such that $ut=w$ and $\ell (u)<\ell (w)$.Template:Sfnp (The name Bruhat order arises from the connection with the Bruhat decomposition of algebraic groups.) Thus, in the symmetric group, one has $u\le w$ if and only if one can get from Template:Mvar to Template:Mvar by applying a sequence of transpositions, each of which swaps two entries Template:Math such that the smaller entry is to the left in the one-line notation.Template:Sfnp The corresponding notion in $S_n^{\pm }$ is as follows: one has that $u<w$ in Bruhat order if there is a sequence of steps $u=u_0\to u_1\to \cdots \to u_k=w$ where at each step we have either $u_{m+1}=u_m\cdot (ij)(\bar{i}\bar{j})$ where Template:Math are elements of $\{\pm 1,\dots ,\pm n\}$ such that $i<j$, $|i|\ne |j|$, and $u_m(i)<u_m(j)$, or $u_{m+1}=u_m\cdot (i\bar{i})$ where $i\in \{1,\dots ,n\}$ is such that $u_m(-i)<u_m(i)$. It follows that, viewing $S_n$ as the subgroup of $S_n^{\pm }$ consisting of signed permutations whose window notation contains no negative entries, one has that two permutations $u,w$ satisfy $u\le w$ as elements of $S_n$ if and only if $u\le w$ as elements of $S_n^{\pm }$.Template:Sfnp

There is another simple criterion to tell whether one permutation is less than another in Bruhat order, in terms of comparing the number of Template:Maths in certain regions of the permutation matrices of Template:Mvar and Template:Mvar.Template:Sfnp The analogous criterion in the hyperoctahedral group is as follows: given a signed permutation Template:Mvar of size Template:Mvar and two integers $i,j\in [-n,n]$, define $v[i,j]$ to be the number of integers $a\in [-n,n]$ such that $a\le i$ and $v(a)\ge j$, where Template:Math is taken to be Template:Math. Then for signed permutations Template:Mvar and Template:Mvar of size Template:Mvar, one has $u\le w$ in Bruhat order if and only if $u[i,j]\le w[i,j]$ for all $i,j\in [-n,n]$. It furthermore follows that $u\le w$ if and only if $\iota (u)$ is less than $\iota (w)$ in the Bruhat order on $S_{2n}$, where Template:Mvar is the inclusion map defined in Template:Slink.Template:Sfnp

Hasse diagrams of the Bruhat order, left weak order, and right weak order for the hyperoctahedral group Template:Math

For any Coxeter group Template:Mvar with simple generating set Template:Mvar, there are two subposets of the Bruhat order called the left and right weak orders.Template:Efn These posets are not the same, but they are isomorphic: an element Template:Mvar is less than Template:Mvar in the right weak order if and only if $u^{-1}$ is less than $w^{-1}$ in the left weak order, so any property of one can be deduced from the corresponding property of the other.Template:Sfnp In general, the left weak order ${\le }_L$ may be characterized in several equivalent ways:

• it is the transitive closure of the relation $u{<}_{L}w$ when there is a simple reflection Template:Mvar in Template:Mvar such that $su=w$ and $\ell (u)<\ell (w)$;
• $u{\le }_{L}w$ if and only if $\ell (u)+\ell (w{u}^{-1})=\ell (w)$; and
• $u{\le }_{L}w$ if and only if the (right) inversion set $T_R(u)=\{t\text{ a reflection in }W:\ell (ut)<\ell (u)\}$ is contained in the right inversion set of Template:Mvar,

among others.Template:SfnpTemplate:SfnpTemplate:SfnpTemplate:Sfnp

In the case of a permutation Template:Mvar in the symmetric group $S_n$, the right inversion set $T_R(w)$ coincides with the notion of inversions described above, namely, a transposition Template:Math is a (right) inversion of Template:Mvar if and only if $i<j$ and $w(i)>w(j)$. In the case of a signed permutation $w\in S_n^{\pm }$, its right inversion set may be decomposed into three pieces, that correspond to the sets

• $\mathrm{Inv}(w)=\{(i,j):1\le i<j\le n\text{ and }w(i)>w(j)\}$,
• $\mathrm{Neg}(w)=\{i:1\le i\le n\text{ and }w(i)<0\}$, and
• $\mathrm{Nsp}(w)=\{(i,j):1\le i<j\le n\text{ and }w(i)+w(j)<0\}$,

and it holds for two signed permutations $u,w$ that $u{\le }_{L}w$ if and only if $\mathrm{Inv}(u)\subseteq \mathrm{Inv}(w)$, $\mathrm{Neg}(u)\subseteq \mathrm{Neg}(w)$, and $\mathrm{Nsp}(u)\subseteq \mathrm{Nsp}(w)$.Template:Sfnp

Absolute order and noncrossing partitions

The six noncrossing partitions of a four-element set that are fixed by 180 degree rotation, with their corresponding elements of the hyperoctahedral group Template:Math

In a reflection group Template:Mvar, the reflection length ${\ell }_T(w)$ of an element Template:Mvar is the smallest number Template:Mvar such that there exist reflections $t_1,\dots ,t_k$ in Template:Mvar such that Template:Mvar is equal to the product $t_1\cdots t_k$ of the reflections. In a finite real reflection group acting on a space Template:Mvar of dimension Template:Mvar, the reflection length of an element Template:Mvar is equal to Template:Math, where $\mathrm{fix}(w)=\{v\in V:w(v)=v\}$ is the fixed space of Template:Mvar.Template:SfnpTemplate:Sfnp In the case that $w\in S_n^{\pm }$ is viewed as a signed permutation, this fixed space dimension is equal to the number of positive cycles of Template:Mvar. That is, if the conjugacy class of Template:Mvar is Template:Math, then the reflection length is $n-\ell (\lambda )$ where $\ell (\lambda )$ is the number of parts of the partition Template:Mvar.Template:Sfnp

In a reflection group Template:Mvar, the absolute order is the partial order ${\le }_T$ on Template:Mvar defined by $u{\le }_{T}w$ if $${\ell }_T(u)+{\ell }_T(u^{-1}w)={\ell }_T(w),$$ and the lattice of $W$-noncrossing partitions is the interval in absolute order that lies below a Coxeter element of Template:Mvar (as in Template:Slink).Template:EfnTemplate:Sfnp This poset derives its name because, in the case of the symmetric group $S_n$, there is a one-to-one correspondence between the elements of the lattice of $S_n$-noncrossing partitions and the set partitions of the set $\{1,2,\dots ,n\}$ that are noncrossing, in the sense that when the numbers Template:Math 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.Template:Sfnp In the case of the hyperoctahedral group $S_n^{\pm }$, the lattice of noncrossing partitions is isomorphic to the sublattice of noncrossing partitions of a Template:Math-element set that are symmetric under Template:Math rotation.Template:SfnpTemplate:SfnpTemplate:Sfnp

The number of noncrossing partitions of an Template:Mvar-element set is the Template:Mvarth Catalan number $C_n=\frac{1}{n+1}(\genfrac{}{}{0ex}{}{2n}{n})$, with the number of partitions at each rank given by a Narayana number.Template:Sfnp The corresponding number of noncrossing partitions for $S_n^{\pm }$ is the central binomial coefficient $(\genfrac{}{}{0ex}{}{2n}{n})$, with the number of elements at rank Template:Mvar equal to ${(\genfrac{}{}{0ex}{}{n}{k})}^2$.Template:SfnpTemplate:Sfnp

Tableau insertion algorithms

Each integer partition $\lambda =⟨{\lambda }_1,\dots ,{\lambda }_{\ell }⟩$ is associated to a Young diagram, an arrangement of left-aligned boxes with ${\lambda }_1$ boxes in the first row, ${\lambda }_2$ boxes in the second row, and so on. A standard Young tableau of shape Template:Mvar is a filling of the Young diagram of Template:Mvar with the numbers Template:Math, Template:Math, ..., $|\lambda |$ so that numbers increase in each row from left to right and in each column from smaller-numbered rows to larger-numbered rows.Template:Sfnp The Robinson–Schensted correspondence is a bijection between the symmetric group $S_n$ and the set of pairs Template:Math of standard Young tableaux such that the shapes of Template:Mvar and Template:Mvar are the same partition of Template:Mvar.Template:Sfnp It provides a crucial connection between combinatorics, the theory of symmetric functions, and the representation theory of the symmetric group.Template:SfnpTemplate:Sfnp

There are several versions of the Robinson–Schensted correspondence for the hyperoctahedral group. In one of these, the role of standard Young tableaux is played by bitableaux: if Template:Mvar and Template:Mvar are two partitions of total size Template:Mvar, a standard bitableau of shape $(\lambda ,\mu )$ is an ordered pair $(T_1,T_2)$ such that $T_1$ and $T_2$ are fillings of the Young diagrams of Template:Mvar and Template:Mvar, respectively, and the two tableaux together are filled with Template:Math and each one obeys the order conditions of a standard Young tableau. Then there is a bijection between $S_n^{\pm }$ and pairs $(P,Q)$ where $P=(P_1,P_2)$ and $Q=(Q_1,Q_2)$ are bitableaux of the same shape (a bipartition of Template:Mvar). The bijection can be realized essentially by applying the usual (symmetric group) Robinson–Schensted correspondence separately to the positive and negative entries of the window notation of a signed permutation.

Associated polyhedra

A truncated cuboctahedron, the three-dimensional type-B permutohedron

In addition to its connection with the hypercube and the cross-polytope, the hyperoctahedral group is associated with other polyhedra. The Template:Mvar-dimensional polytope whose vertices are all the permutations of Template:Math is the type-B permutohedron, a signed analogue of the permutohedron. (A combinatorially equivalent polytope can be constructed by taking the convex hull of the orbit under $S_n^{\pm }$ of any sufficiently generic point in ${ℝ}^n$.)Template:Sfnp Its [[skeleton of a polytope|Template:Math-skeleton]] is the Hasse diagram of the weak order on $S_n^{\pm }$.Template:SfnpTemplate:Sfnp In the case $n=2$ it is an octagon, while in the case $n=3$ it is the truncated cuboctahedron.Template:Sfn

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

Homology

The group homology of the hyperoctahedral group exhibits stabilization, in the sense of stable homotopy theory: the homology groups $H_i(S_n^{\pm })$ and $H_i(S_{n+1}^{\pm })$ are isomorphic whenever $n\ge 2i$, and so in particular they are eventually independent of Template:Mvar. (This is true regardless of the choice of coefficients.)Template:Sfnp

The homologies $H_i(S_n^{\pm })$ with coefficients $\mathbb{Z}$ have been explicitly computed for small Template:Mvar. The first homology group $H_1(S_n^{\pm };\mathbb{Z})$ is the same as the abelianization. For any Coxeter group Template:Mvar, the abelianization is a elementary 2–group with one generator for each component of graph obtained from the Coxeter–Dynkin diagram by deleting all the edges with even labels (equivalently, one generator for each conjugacy class of reflections). Thus, $H_1(S_1^{\pm };\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$ and $H_1(S_n^{\pm };\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ for $n\ge 2$, with stabilization at the Klein four-group.Template:SfnpTemplate:Sfnp

The second homology groups $H_2(S_n^{\pm },\mathbb{Z})$ (also known as the Schur multipliers) were computed by Ihara and Yokonuma. They are trivial for $n=0,1$, and given by $H_2(S_2^{\pm },\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$, $H_2(S_3^{\pm },\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z}{)}^2$, and $H_2(S_n^{\pm },\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z}{)}^3$ for $n\ge 4$. The third homology groups have also been computed; they stabilize at $(\mathbb{Z}/2\mathbb{Z}{)}^7\oplus \mathbb{Z}/3\mathbb{Z}\oplus (\mathbb{Z}/4\mathbb{Z}{)}^2$ for $n\ge 6$.Template:SfnpTemplate:Sfnp

Other properties

The hyperoctahedral group $S_n^{\pm }$ is a subgroup of the orthogonal group $O(n)$ of $n\times n$ orthogonal matrices. It consists exactly of those orthogonal matrices whose entries are all integers.Template:SfnpTemplate:Sfnp

For $n\ge 2$, the group $S_n^{\pm }$ has a minimal generating set of size Template:Math.Template:Sfnp

For any finite group Template:Mvar, the strong symmetric genus of Template:Mvar is the smallest genus of an orientable surface on which Template:Mvar acts faithfully as a group of orientation-preserving automorphisms. For all $n\ge 3$, the strong symmetric genus of $S_n^{\pm }$ is equal to $\frac{n!\cdot 2^{n-3}}{3}+1$, with the exception of $n=5$ (when the genus is Template:Math instead of $\frac{5!\cdot 2^{5-3}}{3}+1=161$), $n=6$ (Template:Math instead of Template:Math), and $n=8$ (Template:Math instead of Template:Math).Template:Sfnp

Generalizations and extensions

Colored permutations

Template:Main The hyperoctahedral groups $S_n^{\pm }$ are generalized by the groups $G(r,n)$ of Template:Mvar-colored permutations, defined as the wreath product $G(r,n)=C_r\wr S_n$ of the cyclic group $C_r$ by the symmetric group $S_n$. The group $G(r,n)$ may be concretely represented as the group of $n\times n$ monomial matrices whose nonzero entries are complex Template:Mvarth roots of unity. For $r>2$, these groups are no longer real reflection groups; instead, they belong to the infinite family of imprimitive complex reflection groups.Template:Sfnp In particular, the group $G(r,n)$ is generated by Template:Mvar complex reflections: the Template:Math matrices of the adjacent transpositions, together with the matrix $$\left[\begin{array}{cccc}\zeta & 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right],$$ where $\zeta =e^{2\pi i/r}$ is a primitive Template:Mvarth root of unity.Template:Sfnp

Affine Coxeter groups

The fixed lines of the reflections of the affine group ${\tilde{S}}_2^{\pm }$ divide the plane into isosceles right triangles.

Each finite Weyl group Template:Mvar is associated to an affine Coxeter group, which is generated by the group Template:Mvar together with a reflection across a translated copy of one of the reflecting hyperplanes of a reflection in Template:Mvar. In the case of the dihedral group $S_2^{\pm }$ of order Template:Math, 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).Template:SfnpTemplate:Sfnp When $n>2$, there are two different affine extensions of $S_n^{\pm }$, corresponding to the two different crystallographic root systems associated to the group. Viewing $S_n^{\pm }$ as a Weyl group acting by linear transformations on ${ℝ}^n$, in both cases, the affine group is generated by the finite group $S_n^{\pm }$ together with a single reflection, across a hyperplane with equation $x\cdot \alpha =1$ where $\alpha$ is the highest root of the root system and Template:Math is the standard dot product. In type B, the highest root is $e_{n-1}+e_n$, corresponding to the reflection across the hyperplane $x_{n-1}+x_n=1$, while in type C, the highest root is $2e_n$, corresponding to the reflection across the hyperplane with equation $2x_n=1$.Template:SfnpTemplate:Sfnp

The affine type B and C Coxeter groups associated with $S_n^{\pm }$ have Coxeter–Dynkin diagrams

Template:Math and Template:Math,

respectively.Template:SfnpTemplate:Sfnp

Both affine groups also have combinatorial realizations. The affine group of type C can be realized as the set of bijections $w:\mathbb{Z}\to \mathbb{Z}$ such that $w(-i)=-w(i)$ and $w(2n+2-i)=2n+2-w(i)$ for all $i\in \mathbb{Z}$. In other words, these are the bijections on $\mathbb{Z}$ that commute with the reflection across Template:Math and also the reflection across Template:Math (and consequently with the infinite dihedral group of symmetries of $\mathbb{Z}$ generated by these two reflections). By construction, all permutations in this combinatorial model fix the multiples of Template:Math. Alternatively, one may consider a slightly different combinatorial model, consisting of permutations $w:\mathbb{Z}\to \mathbb{Z}$ such that $w(-i)=-w(i)$ and $w(i+2n+1)=w(i)+2n+1$ for all $i\in \mathbb{Z}$. In either model, such a permutation is completely determined by its Template:Mvar values $w(1),\dots ,w(n)$.Template:SfnpTemplate:Sfnp

For $n\ge 3$, 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 Template:Mvar such that Template:Math is negative is even. (This number is always finite.)Template:SfnpTemplate:Sfnp Both affine groups can furthermore be realized as subgroups of an appropriate affine symmetric group.Template:Sfnp

Braid group

Each Coxeter group Template:Mvar is associated to an Artin–Tits group $B_W$, which is defined by a similar presentation that omits relations of the form $s^2=1$ for each generator Template:Mvar.Template:Sfnp In particular, the Artin–Tits group associated to $S_n^{\pm }$ is generated by Template:Mvar elements ${\sigma }_0,{\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to the relations ${\sigma }_0{\sigma }_1{\sigma }_0{\sigma }_1={\sigma }_1{\sigma }_0{\sigma }_1{\sigma }_0$ and ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$ for $i=1,\dots ,n-2$ (and no others).Template:Sfnp

Artin–Tits groups are sometimes also known as generalized braid groups, because the Artin–Tits group $B_{S_n}$ of the (finite) symmetric group is the braid group on Template:Mvar strands.Template:Sfnp Not all Artin–Tits groups have a natural representation in terms of geometric braids. However, the Artin–Tits group of the hyperoctahedral group $S_n^{\pm }$ does have such a representation: it is given by the subgroup of the braid group on $n+1$ 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 Template:Mvar strands in an annular region.Template:SfnpTemplate:Sfnp The Artin–Tits group of $S_n^{\pm }$ is also isomorphic to the mapping class group of a closed disc with Template:Math marked points, with a single point fixed pointwise,Template:Sfnp and can be written as a semidirect product of $B_{{\tilde{S}}_n}$ with an infinite cyclic group.Template:Sfnp

Infinite hyperoctahedral group

If one views the hyperoctahedral group $S_n^{\pm }$ as acting on $\mathbb{Z}$, fixing all integers of magnitude larger than Template:Mvar, this induces a containment $S_1^{\pm }\subset S_2^{\pm }\subset S_3^{\pm }\subset \cdots$. Then the infinite hyperoctahedral group $S_{\mathrm{\infty }}^{\pm }$ can be defined as the union $S_{\mathrm{\infty }}^{\pm }=\bigcup _{n\ge 1}{S}_n^{\pm }$ of these groups (or in other words as the direct limit of the sequence of inclusions).Template:SfnpTemplate:Sfnp Combinatorially, this group may be viewed as the group of bijections $w:\mathbb{Z}\to \mathbb{Z}$ such that $w(-i)=-w(i)$ for all Template:Mvar and $w(i)\ne i$ for only finitely many $i\in \mathbb{Z}$.Template:Sfnp

Tables and diagrams

The Coxeter–Dynkin diagram for $S_n^{\pm }$ is

Template:Math

with Template:Mvar nodes;Template:SfnpTemplate:Sfnp the corresponding Coxeter bracket notation is

Template:Math.

The Coxeter–Dynkin diagrams of the associated affine Coxeter groups are

Template:Math

for type ${\tilde{\mathrm{B}}}_n$ and

Template:Math

for type ${\tilde{\mathrm{C}}}_n$.Template:SfnpTemplate:Sfnp

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

	$S_n^{\pm }$
CardinalityTemplate:Sfnp	$2^{n}n!$
DegreesTemplate:Sfnp	Template:Math
Coxeter number Template:MvarTemplate:Sfnp	Template:Math
Number of reflectionsTemplate:Sfnp	$n^2$
Number of positive rootsTemplate:Sfnp	$n^2$
Number of rootsTemplate:Sfnp	$2n^2$
Exponents	Template:Math
Template:Mvar-Catalan numberTemplate:Sfnp	$(\genfrac{}{}{0ex}{}{2n}{n})$
Template:Mvar-Narayana numbersTemplate:Sfnp	${(\genfrac{}{}{0ex}{}{n}{k})}^2$, $k=0,1,\dots ,n$
Connection indexTemplate:EfnTemplate:SfnpTemplate:Sfnp	Template:Math

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

Works cited

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