Hyperoctahedral group

Last updated February 24, 2024

The $C 2$ group has order 8 as shown on this circle	The $C 3$ ( $O h$ ) group has order 48 as shown by these spherical triangle reflection domains.

In mathematics, a hyperoctahedral group is an important type of group that can be realized 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 $n$ , the dimension of the hypercube.

As a Coxeter group it is of type $B n = C n$ , 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 $S n$ is the symmetric group of degree $n$ . As a permutation group, the group is the signed symmetric group of permutations π either of the set $\{-n,-n+1,\cdots ,-1,1,2,\cdots ,n\}$ or of the set $\{-n,-n+1,\cdots ,n\}$ such that $\pi (i)=-\pi (-i)$ for all $i$ . As a matrix group, it can be described as the group of $n \times n$ orthogonal matrices whose entries are all integers. Equivalently, this is the set of $n \times n$ 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 ( Young 1930 ) according to ( Kerber 1971 , p. 2).

In three dimensions, the hyperoctahedral group is known as $O \times S 2$ where $O ≅ S 4$ is the octahedral group, and $S 2$ 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

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

n	Symmetry group	B_n	Coxeter notation		Order	Mirrors	Structure	Related regular polytopes
2	D₄ (*4•)	B₂	[4]		2²2! = 8	4	$Dih_{4}$ $\cong S_{2}\wr S_{2}$	Square, octagon
3	O_h (*432)	B₃	[4,3]		2³3! = 48	3+6	$S_{4}\times S_{2}$ $\cong S_{2}\wr S_{3}$	Cube, octahedron
4	±¹/₆[OxO].2 ^[1] (O/V;O/V)^*^[2]	B₄	[4,3,3]		2⁴4! = 384	4+12	$S_{2}\wr S_{4}$	Tesseract, 16-cell, 24-cell
5		B₅	[4,3,3,3]		2⁵5! = 3840	5+20	$S_{2}\wr S_{5}$	5-cube, 5-orthoplex
6		B₆	[4,3⁴]		2⁶6! = 46080	6+30	$S_{2}\wr S_{6}$	6-cube, 6-orthoplex
...n		B_n	[4,3^n-2]	...	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.

The hyperoctahedral subgroup, D_n by dimension:

n	Symmetry group	D_n	Coxeter notation		Order	Mirrors	Related polytopes
2	D₂ (*2•)	D₂	[2] = [ ]×[ ]		4	2	Rectangle
3	T_d (*332)	D₃	[3,3]		24	6	tetrahedron
4	±¹/₃[TxT].2 ^[1] (T/V;T/V)₋^*^[3]	D₄	[3^1,1,1]		192	12	16-cell
5		D₅	[3^2,1,1]		1920	20	5-demicube
6		D₆	[3^3,1,1]		23040	30	6-demicube
...n		D_n	[3^n-3,1,1]	...	2^n-1n!	n(n-1)	demihypercube

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]⁺		4
3	O (432)	[4,3]⁺		24
4	¹/₆[O×O].2 ^[1] (O/V;O/V) ^[4]	[4,3,3]⁺		192
5		[4,3,3,3]⁺		1920
6		[4,3,3,3,3]⁺		23040
...n		[4,(3^n-2)⁺]	...	2^n-1n!

Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:^[5] These groups have n orthogonal mirrors in n-dimensions.

n	Symmetry group	Coxeter notation		Order	Mirrors	Related polytopes
2	D₂ (*2•)	[4,1⁺]=[2]		4	2	Rectangle
3	T_h (3*2)	[4,3⁺]		24	3	snub octahedron
4	±¹/₃[T×T].2 ^[1] (T/V;T/V)^*^[6]	[4,(3,3)⁺]		192	4	snub 24-cell
5		[4,(3,3,3)⁺]		1920	5
6		[4,(3,3,3,3)⁺]		23040	6
...n		[4,(3^n-2)⁺]	...	2^n-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},\mathbf {Z} )={\begin{cases}0&n=0\\\mathbf {Z} /2&n=1\\\mathbf {Z} /2\times \mathbf {Z} /2&n\geq 2\end{cases}}.

This is easily seen directly: the $-1$ elements are order 2 (which is non-empty for $n\geq 1$ ), and all conjugate, as are the transpositions in $S_{n}$ (which is non-empty for $n\geq 2$ ), 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 ( Ihara & Yokonuma 1965 ).

They are:

H_{2}(C_{n},\mathbf {Z} )={\begin{cases}0&n=0,1\\\mathbf {Z} /2&n=2\\(\mathbf {Z} /2)^{2}&n=3\\(\mathbf {Z} /2)^{3}&n\geq 4\end{cases}}.

Notes

1 2 3 4 Conway & Smith 2003
↑ du Val 1964 , #47
↑ du Val 1964 , #42
↑ du Val 1964 , #27
↑ Coxeter 1999 , p. 121, Essay 5 Regular skew polyhedra
↑ du Val 1964 , #41

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References

Miller, G. A. (1918). "Groups formed by special matrices". Bull. Am. Math. Soc. 24 (4): 203–6. doi: 10.1090/S0002-9904-1918-03043-7 .
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Kerber, Adalbert (1971), Representations of permutation groups. I, Lecture Notes in Mathematics, vol. 240, Springer-Verlag, doi:10.1007/BFb0067943, ISBN 978-3-540-05693-5, MR 0325752
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[Conway03-1] 1 2 3 4 Conway & Smith 2003

[2] u Val 1964 , #47

[3] u Val 1964 , #42

[4] u Val 1964 , #27

[5] Coxeter 1999 , p. 121, Essay 5 Regular skew polyhedra

[6] u Val 1964 , #41

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