BREAKING NEWS
Hyperoctahedral group

## Summary

 The C2 group has order 8 as shown on this circle The C3 (Oh) 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 Bn = Cn, 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 ${\displaystyle S_{2}\wr S_{n}}$ where Sn 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 ${\displaystyle \{-n,-n+1,\cdots ,-1,1,2,\cdots ,n\}}$ or of the set ${\displaystyle \{-n,-n+1,\cdots ,n\}}$ such that ${\displaystyle \pi (i)=-\pi (-i)}$ for all i. As a matrix group, it can be described as the group of n × n orthogonal matrices whose entries are all integers. Equivalently, this is the set of n × 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 × S2 where OS4 is the octahedral group, and S2 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 Bn, 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]     222! = 8 4 ${\displaystyle Dih_{4}}$  ${\displaystyle \cong S_{2}\wr S_{2}}$  Square, octagon
3 Oh (*432) B3 [4,3]       233! = 48 3+6 ${\displaystyle S_{4}\times S_{2}}$  ${\displaystyle \cong S_{2}\wr S_{3}}$  Cube, octahedron
4 ±1/6[OxO].2 [1]
(O/V;O/V)* [2]
B4 [4,3,3]         244! = 384 4+12 ${\displaystyle S_{2}\wr S_{4}}$  Tesseract, 16-cell, 24-cell
5   B5 [4,3,3,3]           255! = 3840 5+20 ${\displaystyle S_{2}\wr S_{5}}$  5-cube, 5-orthoplex
6   B6 [4,34]             266! = 46080 6+30 ${\displaystyle S_{2}\wr S_{6}}$  6-cube, 6-orthoplex
...n   Bn [4,3n-2]       ...     2nn! = (2n)!! n2 ${\displaystyle S_{2}\wr S_{n}}$  hypercube, orthoplex

## Subgroups

There is a notable index two subgroup, corresponding to the Coxeter group Dn 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 ${\displaystyle \{\pm 1\}}$ ), and one map coming from the parity of the permutation. Multiplying these together yields a third map ${\displaystyle C_{n}\to \{\pm 1\}}$ . The kernel of the first map is the Coxeter group ${\displaystyle D_{n}.}$  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, Dn by dimension:

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

${\displaystyle H_{1}(C_{n},\mathbf {Z} )={\begin{cases}0&n=0\\\mathbf {Z} /2&n=1\\\mathbf {Z} /2\times \mathbf {Z} /2&n\geq 2\end{cases}}.}$

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

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

## Notes

1. ^ a b c d Conway & Smith 2003
2. ^ du Val 1964, #47
3. ^ du Val 1964, #42
4. ^ du Val 1964, #27
5. ^ Coxeter 1999, p. 121, Essay 5 Regular skew polyhedra
6. ^ du Val 1964, #41

## 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.
• du Val, P. (1964). Homographies, Quaternions and Rotations. Oxford mathematical monographs. Clarendon Press. OCLC 904102141.
• Ihara, Shin-ichiro; Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of finite reflection groups", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 11: 155–171, ISSN 0040-8980, MR 0190232
• 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
• Kerber, Adalbert (1975), Representations of permutation groups. II, Lecture Notes in Mathematics, vol. 495, Springer-Verlag, doi:10.1007/BFb0085740, ISBN 978-3-540-07535-6, MR 0409624
• Young, Alfred (1930), "On Quantitative Substitutional Analysis 5", Proceedings of the London Mathematical Society, Series 2, 31: 273–288, doi:10.1112/plms/s2-31.1.273, ISSN 0024-6115, JFM 56.0135.02
• Coxeter, H.S.M.; Moser, W.O.J. (2013) [1980]. Generators and Relations for Discrete Groups (4th ed.). Springer. p. 92 §7.4 Linear fractional groups, p. 122 §9.3 Finite Groups. ISBN 978-3-662-21943-0.
• Baake, M. (1984). "Structure and representations of the hyperoctahedral group". J. Math. Phys. 25 (11): 3171. Bibcode:1984JMP....25.3171B. doi:10.1063/1.526087.
• Stembridge, John R. (1992). "The projective representations of the hyperoctahedral group". J. Algebra. 145 (2): 396–453. doi:10.1016/0021-8693(92)90110-8. hdl:2027.42/30235.
• Coxeter, H.S.M. (1999). The Beauty of Geometry: Twelve Essays. Dover. ISBN 0-486-40919-8. LCCN 99035678.
• Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. CRC Press. ISBN 978-1-000-68777-4.