Hyperoctahedral group

The Template:Math group has order 8 as shown on this circle

The Template:Math (Template:Math) group has order 48 as shown by these spherical triangle reflection domains.

A hyperoctahedral group is a type of mathematical group that arises 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 Template:Mvar, the dimension of the hypercube.

As a Coxeter group it is of type Template:Math, and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is $S_2\wr S_n$ where Template:Mvar is the symmetric group of degree Template:Mvar. As a permutation group, the group is the signed symmetric group of permutations π either of the set Template:Tmath or of the set Template:Tmath such that Template:Tmath for all Template:Mvar. As a matrix group, it can be described as the group of Template:Math orthogonal matrices whose entries are all integers. Equivalently, this is the set of Template:Math matrices with entries only 0, 1, or –1, which are invertible, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral group was described by Template:Harv according to Template:Harv.

In three dimensions, the hyperoctahedral group is known as Template:Math where Template:Math is the octahedral group, and Template:Math is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex. In 4-dimensions it is called a hexadecachoric symmetry, after the regular 16-cell, or 4-orthoplex. In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.

By dimension

The 8 permutations of the square, forming D₄

8 of the 48 permutations of a cube, forming O_h

Hyperoctahedral groups can be named as B_n, a bracket notation, or as a Coxeter group graph:

n	Symmetry group	Bn	Coxeter notation		Order	Mirrors	Structure	Related regular polytopes
2	D4 (*4•)	B2	[4]	Template:CDD	222! = 8	4	$Di{h}_4$ $\cong S_2\wr S_2$	Square, octagon
3	Oh (*432)	B3	[4,3]	Template:CDD	233! = 48	3+6	$S_4\times S_2$ $\cong S_2\wr S_3$	Cube, octahedron
4	±1/6[OxO].2 (O/V;O/V)*	B4	[4,3,3]	Template:CDD	244! = 384	4+12	$S_2\wr S_4$	Tesseract, 16-cell, 24-cell
5		B5	[4,3,3,3]	Template:CDD	255! = 3840	5+20	$S_2\wr S_5$	5-cube, 5-orthoplex
6		B6	[4,34]	Template:CDD	266! = 46080	6+30	$S_2\wr S_6$	6-cube, 6-orthoplex
...n		Bn	[4,3n-2]	Template:CDD...Template:CDD	2nn! = (2n)!!	n2	$S_2\wr S_n$	hypercube, orthoplex

Subgroups

There is a notable index two subgroup, corresponding to the Coxeter group D_n and the symmetries of the demihypercube. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the n copies of $\{\pm 1\}$), and one map coming from the parity of the permutation. Multiplying these together yields a third map $C_n\to \{\pm 1\}$. The kernel of the first map is the Coxeter group $D_n.$ In terms of signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.

The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in H₁: Abelianization below, and their intersection is the derived subgroup, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.

In the other direction, the center is the subgroup of scalar matrices, {±1}; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group.

In dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih₄ of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2). In general, passing to the subquotient (derived subgroup, mod center) is the symmetry group of the projective demihypercube.

Tetrahedral symmetry in three dimensions, order 24

The hyperoctahedral subgroup, D_n by dimension:

n	Symmetry group	Dn	Coxeter notation		Order	Mirrors	Related polytopes
2	D2 (*2•)	D2	[2] = [ ]×[ ]	Template:CDD	4	2	Rectangle
3	Td (*332)	D3	[3,3]	Template:CDD	24	6	tetrahedron
4	±1/3[TxTemplate:Overline].2 (T/V;T/V)−*	D4	[31,1,1]	Template:CDD	192	12	16-cell
5		D5	[32,1,1]	Template:CDD	1920	20	5-demicube
6		D6	[33,1,1]	Template:CDD	23040	30	6-demicube
...n		Dn	[3n-3,1,1]	Template:CDD...Template:CDD	2n-1n!	n(n-1)	demihypercube

Pyritohedral symmetry in three dimensions, order 24

Octahedral symmetry in three dimensions, order 24

The chiral hyper-octahedral symmetry, is the direct subgroup, index 2 of hyper-octahedral symmetry.

n	Symmetry group	Coxeter notation		Order
2	C4 (4•)	[4]+	Template:CDD	4
3	O (432)	[4,3]+	Template:CDD	24
4	1/6[O×O].2 (O/V;O/V)	[4,3,3]+	Template:CDD	192
5		[4,3,3,3]+	Template:CDD	1920
6		[4,3,3,3,3]+	Template:CDD	23040
...n		[4,(3n-2)+]	Template:CDD...Template:CDD	2n-1n!

Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension: These groups have n orthogonal mirrors in n-dimensions.

n	Symmetry group	Coxeter notation		Order	Mirrors	Related polytopes
2	D2 (*2•)	[4,1+]=[2]	Template:CDD	4	2	Rectangle
3	Th (3*2)	[4,3+]	Template:CDD	24	3	snub octahedron
4	±1/3[T×T].2 (T/V;T/V)*	[4,(3,3)+]	Template:CDD	192	4	snub 24-cell
5		[4,(3,3,3)+]	Template:CDD	1920	5
6		[4,(3,3,3,3)+]	Template:CDD	23040	6
...n		[4,(3n-2)+]	Template:CDD...Template:CDD	2n-1n!	n

Homology

The group homology of the hyperoctahedral group is similar to that of the symmetric group, and exhibits stabilization, in the sense of stable homotopy theory.

H₁: abelianization

The first homology group, which agrees with the abelianization, stabilizes at the Klein four-group, and is given by:

$H_1(C_n,𝐙)=\{\begin{array}{cc}0& n=0\\ 𝐙/2& n=1\\ 𝐙/2\times 𝐙/2& n\ge 2\end{array}.$

This is easily seen directly: the $-1$ elements are order 2 (which is non-empty for $n\ge 1$), and all conjugate, as are the transpositions in $S_n$ (which is non-empty for $n\ge 2$), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to $-1\in \{\pm 1\},$ as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the n copies of $\{\pm 1\}$), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the determinant of the matrix, which sends both these classes to $-1$), and together with the trivial map these form the 4-group.

H₂: Schur multipliers

The second homology groups, known classically as the Schur multipliers, were computed in Template:Harv.

They are:

$H_2(C_n,𝐙)=\{\begin{array}{cc}0& n=0,1\\ 𝐙/2& n=2\\ (𝐙/2{)}^2& n=3\\ (𝐙/2{)}^3& n\ge 4\end{array}.$

References

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