View source for Hyperoctahedral group

{| class=wikitable align=right width=240
|[[File:C2 group circle domains.png|120px]]<BR>The C<sub>2</sub> group has order 8 as shown on this circle
|[[File:Sphere symmetry group oh.png|120px]]<BR>The C<sub>3</sub> (O<sub>h</sub>) group has order 48 as shown by these [[Möbius triangles|spherical triangle reflection domains]].
|}
In [[mathematics]], a '''hyperoctahedral group''' is an important type of group that can be realized as the [[symmetry group|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<sub>''n''</sub> = C<sub>''n''</sub>, and as a [[Weyl group]] it is associated to the [[orthogonal group]]s in odd dimensions.  As a [[wreath product]] it is <math>S_2 \wr S_n</math> where <math>S_n</math> is the [[symmetric group]] of degree ''n''.  As a [[permutation group]], the group is the '''signed symmetric group''' of permutations&nbsp;''π'' either of the set { −''n'', −''n''&nbsp;+&nbsp;1, ..., −1, 1, 2, ..., ''n'' } or of the set { −''n'', −''n''&nbsp;+&nbsp;1, ..., ''n'' } such that ''π''(''i'') = −''π''(−''i'') for all&nbsp;''i''.  As a [[matrix group]], it can be described as the group of ''n''×''n'' [[orthogonal matrices]] whose entries are all [[integer]]s. The representation theory of the hyperoctahedral group was described by {{harv|Young|1930}} according to {{harv|Kerber|1971|p=2}}.

In three dimensions, the hyperoctahedral group is known as ''O''×''S''<sub>2</sub> where ''O''≅''S''<sub>4</sub> is the [[octahedral group]], and ''S''<sub>2</sub> is a symmetric group (equivalently, [[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 [[Point groups in four dimensions#Hexadecachoric symmetry|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 ==

[[File:Hyperoctahedral group 2; passive prefix.svg|thumb|The 8 permutations of the square, forming D<sub>4</sub>]]
[[File:3-ary Boolean functions; cube permutations; 5.svg|thumb|8 of the 48 permutations of a cube, forming O<sub>h</sub>]]
Hyperoctahedral groups can be named as [[Coxeter group#Properties|B<sub>n</sub>]], a bracket notation, or as a Coxeter group graph:<!-- In the table of symmetry groups, all the entries in the column labeled "Coxeter Notation" have the error "Script Error: No such module 'CDD' ". Unfortunately, I don't know how to fix this. -->

{| class="wikitable"
!n
!Symmetry<BR>group
!B<sub>n</sub>
!colspan=2|[[Coxeter notation]]
![[Symmetry order|Order]]
!Mirrors
!Structure
!Related [[regular polytope]]s
|- align=center
!2
|[[Examples of groups#dihedral group of order 8|D<sub>4</sub>]] (*4&bull;)
|B<sub>2</sub>||[4]||{{CDD|node|4|node}}||2<sup>2</sup>2! = 8||4
| <math>Dih_4</math> <math>\cong S_2 \wr S_2</math>
|[[square (geometry)|Square]], [[octagon]]
|- align=center
!3
|[[Octahedral symmetry|O<sub>h</sub>]] ([[Orbifold notation|*432]])
|B<sub>3</sub>||[4,3]||{{CDD|node|4|node|3|node}}||2<sup>3</sup>3! = [[48 (number)|48]]||3+6
| <math>S_4 \times S_2</math> <math>\cong S_2 \wr S_3</math>
|[[Cube]], [[octahedron]]
|- align=center
!4
|±<sup>1</sup>/<sub>6</sub>[OxO].2 <ref>Conway, 2003</ref><BR>(O/V;O/V)<sup>*</sup> <ref>Du Val, 1964, #47</ref>
|B<sub>4</sub>||[4,3,3]||{{CDD|node|4|node|3|node|3|node}}||2<sup>4</sup>4! = [[384 (number)|384]]||4+12
| <math>S_2 \wr S_4</math>
|[[Tesseract]], [[16-cell]], [[24-cell]]
|- align=center
!5
|&nbsp;
|B<sub>5</sub>||[4,3,3,3]||{{CDD|node|4|node|3|node|3|node|3|node}}||2<sup>5</sup>5! = 3840||5+20
| <math>S_2 \wr S_5</math>
|[[5-cube]], [[5-orthoplex]]
|- align=center
!6
|&nbsp;
|B<sub>6</sub>||[4,3<sup>4</sup>]||{{CDD|node|4|node|3|node|3|node|3|node|3|node}}||2<sup>6</sup>6! = 46080||6+30
| <math>S_2 \wr S_6</math>
|[[6-cube]], [[6-orthoplex]]
|- align=center
!...
|- align=center
!n
|&nbsp;
|B<sub>''n''</sub>||[4,3<sup>n-2</sup>]||{{CDD|node|4|node|3|node|3}}...{{CDD|3|node|3|node}}||2<sup>''n''</sup>''n''!&nbsp;=&nbsp;[[double factorial|(2''n'')!!]]||n<sup>2</sup>
| <math>S_2 \wr S_n</math>
|[[hypercube]], [[orthoplex]]
|}

==Subgroups==
There is a notable index two subgroup, corresponding to the Coxeter group ''D''<sub>''n''</sub> 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 <math>\{\pm 1\}</math>), and one map coming from the parity of the permutation. Multiplying these together yields a third map <math>C_n \to \{\pm 1\}</math>. The kernel of the first map is the Coxeter group <math>D_n.</math> 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|''H''<sub>1</sub>: 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 of order 8|dihedral group Dih<sub>4</sub> 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.
[[File:Sphere symmetry group td.png|thumb|[[Tetrahedral symmetry]] in three dimensions, order 24]]
The '''hyperoctahedral''' subgroup, D<sub>n</sub> by dimension:

{| class="wikitable"
!n
!Symmetry<BR>group
!D<sub>n</sub>
!colspan=2|[[Coxeter notation]]
![[Symmetry order|Order]]
!Mirrors
!Related polytopes
|- align=center
!2
|[[Dihedral symmetry|D<sub>2</sub>]] (*2&bull;)
|D<sub>2</sub>||[2] = [ ]×[ ]||{{CDD|nodes}}||4||2
|[[Rectangle]]
|- align=center
!3
|[[Tetrahedral symmetry|T<sub>d</sub>]] ([[Orbifold notation|*332]])
|D<sub>3</sub>||[3,3]||{{CDD|node|split1|nodes}}||24||6
|[[tetrahedron]]
|- align=center
!4
|±<sup>1</sup>/<sub>3</sub>[Tx{{Overline|T}}].2 <ref>Conway, 2003</ref><BR>(T/V;T/V)<sub>−</sub><sup>*</sup> <ref>Du Val, 1964, #42</ref>
|D<sub>4</sub>||[3<sup>1,1,1</sup>]||{{CDD|node|3|node|split1|nodes}}||192||12
|[[16-cell]]
|- align=center
!5
|&nbsp;
|D<sub>5</sub>||[3<sup>2,1,1</sup>]||{{CDD|node|3|node|3|node|split1|nodes}}||1920||20
|[[5-demicube]]
|- align=center
!6
|&nbsp;
|D<sub>6</sub>||[3<sup>3,1,1</sup>]||{{CDD|node|3|node|3|node|3|node|split1|nodes}}||23040||30
|[[6-demicube]]
|- align=center
!...n
|&nbsp;
|D<sub>n</sub>||[3<sup>n-3,1,1</sup>]||{{CDD|node|3|node}}...{{CDD|3|node|split1|nodes}}||2<sup>n-1</sup>n!||n(n-1)
|[[demihypercube]]
|}
[[File:Sphere symmetry group th.png|thumb|[[Pyritohedral symmetry]] in three dimensions, order 24]]

[[File:Sphere symmetry group o.png|thumb|[[Octahedral symmetry]] in three dimensions, order 24]]

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

{| class="wikitable"
!n
!Symmetry<BR>group
!colspan=2|[[Coxeter notation]]
![[Symmetry order|Order]]
|- align=center
!2
|[[Cyclic symmetry|C<sub>4</sub>]] (4&bull;)
||[4]<sup>+</sup>||{{CDD|node_h2|4|node_h2}}||4
|- align=center
!3
|[[Octahedral symmetry|O]] ([[Orbifold notation|432]])
||[4,3]<sup>+</sup>||{{CDD|node_h2|4|node_h2|3|node_h2}} ||24
|- align=center
!4
|<sup>1</sup>/<sub>6</sub>[O×O].2 <ref>Conway, 2003</ref><BR>(O/V;O/V) <ref>Du Val, 1964, #27</ref>
||[4,3,3]<sup>+</sup>||{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2}} ||192
|- align=center
!5
|&nbsp;
||[4,3,3,3]<sup>+</sup>||{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2|3|node_h2}}||1920
|- align=center
!6
|&nbsp;
||[4,3,3,3,3]<sup>+</sup>||{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2|3|node_h2|3|node_h2}}||23040
|- align=center
!...n
|&nbsp;
||[4,(3<sup>n-2</sup>)<sup>+</sup>]||{{CDD|node_h2|4|node_h2|3|node_h2}}...{{CDD|3|node_h2|3|node_h2}}||2<sup>n-1</sup>n!
|}

Another notable index 2 subgroup can be called '''hyper-pyritohedral symmetry''', by dimension:<ref>Coxeter (1999), p.121, Essay 5 ''Regular skew polyhedra''</ref> These groups have ''n'' orthogonal mirrors in ''n''-dimensions.

{| class="wikitable"
!n
!Symmetry<BR>group
!colspan=2|[[Coxeter notation]]
![[Symmetry order|Order]]
!Mirrors
!Related polytopes
|- align=center
!2
|[[Dihedral symmetry|D<sub>2</sub>]] (*2&bull;)
||[4,1<sup>+</sup>]=[2]||{{CDD|node|4|node_h2}}||4||2
|[[Rectangle]]
|- align=center
!3
|[[Pyritohedral symmetry|T<sub>h</sub>]] ([[Orbifold notation|3*2]])
||[4,3<sup>+</sup>]||{{CDD|node|4|node_h2|3|node_h2}} ||24||3
|[[Icosahedron#Uniform colorings and subsymmetries|snub octahedron]]
|- align=center
!4
|±<sup>1</sup>/<sub>3</sub>[T×T].2 <ref>Conway, 2003</ref><BR>(T/V;T/V)<sup>*</sup> <ref>Du Val, 1964, #41</ref>
||[4,(3,3)<sup>+</sup>]||{{CDD|node|4|node_h2|3|node_h2|3|node_h2}} ||192||4
|[[snub 24-cell]]
|- align=center
!5
|&nbsp;
||[4,(3,3,3)<sup>+</sup>]||{{CDD|node|4|node_h2|3|node_h2|3|node_h2|3|node_h2}}||1920||5
|
|- align=center
!6
|&nbsp;
||[4,(3,3,3,3)<sup>+</sup>]||{{CDD|node|4|node_h2|3|node_h2|3|node_h2|3|node_h2|3|node_h2}}||23040||6
|
|- align=center
!...n
|&nbsp;
||[4,(3<sup>n-2</sup>)<sup>+</sup>]||{{CDD|node|4|node_h2|3|node_h2}}...{{CDD|3|node_h2|3|node_h2}}||2<sup>n-1</sup>n!||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]].

===H<sub>1</sub>: abelianization===
The first homology group, which agrees with the [[abelianization]], stabilizes at the [[Klein four-group]], and is given by:
:<math>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}.</math>
This is easily seen directly: the <math>-1</math> elements are order 2 (which is non-empty for <math>n\geq 1</math>), and all conjugate, as are the transpositions in <math>S_n</math> (which is non-empty for <math>n\geq 2</math>), 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 <math>-1 \in \{\pm 1\},</math> 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 <math>\{\pm 1\}</math>), 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 <math>-1</math>), and together with the trivial map these form the 4-group.

===H<sub>2</sub>: Schur multipliers===
The second homology groups, known classically as the [[Schur multiplier]]s, were computed in {{Harv|Ihara|Yokonuma|1965}}.

They are:
:<math>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}.</math>

==Notes==
{{reflist}}

== References ==
{{refbegin}}
* {{cite article| first1=G. A.|last1=Miller| year=1918 | doi=10.1090/S0002-9904-1918-03043-7 
  |title=Groups formed by special matrices| journal = Bull. Am. Math. Soc. | pages = 203–206| volume=24}}
* [[Patrick du Val]], ''Homographies, Quaternions and Rotations'' (1964)
* {{Citation | last1=Ihara | first1=Shin-ichiro | last2=Yokonuma | first2=Takeo | title=On the second cohomology groups (Schur-multipliers) of finite reflection groups | mr=0190232 | year=1965 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=11 | pages=155–171}}
*{{Citation | last1=Kerber | first1=Adalbert | title=Representations of permutation groups. I | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0067943 | mr=0325752 | year=1971 | volume=240| isbn=978-3-540-05693-5 }}
*{{Citation | last1=Kerber | first1=Adalbert | title=Representations of permutation groups. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics| doi=10.1007/BFb0085740 | mr=0409624 | year=1975 | volume=495| isbn=978-3-540-07535-6 }}
*{{Citation | last1=Young | first1=Alfred | author1-link=Alfred Young | title=On Quantitative Substitutional Analysis 5 | doi=10.1112/plms/s2-31.1.273 | jfm=56.0135.02 | year=1930 | journal=Proceedings of the London Mathematical Society|series=Series 2 | issn=0024-6115 | volume=31 | pages=273–288| url=https://zenodo.org/record/1447746 }}
* H.S.M. Coxeter and W. O. J. Moser. ''Generators and Relations for Discrete Groups'' 4th ed, Springer-Verlag. New York. 1980 p92, p122
* {{cite article|first1= M. | last1=Baake | title = Structure and representations of the hyperoctahedral group
  |year=1984 | doi= 10.1063/1.526087 | journal = J. Math. Phys. | volume =25 | page = 3171}}
* {{cite article|first1= John R. | last1=Stembridge | doi= 10.1016/0021-8693(92)90110-8
  |title= The projective representations of the hyperoctahedral group| journal =J. Algebra | year = 1992
   |pages=396–453 | volume=145 | number=2}}
* [[Coxeter]], ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{ISBN|0-486-40919-8}}
* [[John Horton Conway]], ''On Quaternions and Octonions'' (2003)
{{refend}}

[[Category:Finite reflection groups]]