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Hyperoctahedral group#By dimension

{{Short description|Group of symmetries of an n-dimensional hypercube}}

class=wikitable align=right width=240

|120px
The {{math|C{{sub|2}}}} group has order 8 as shown on this circle

|120px
The {{math|C{{sub|3}}}} ({{math|O{{sub|h}}}}) 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 {{mvar|n}}, the dimension of the hypercube.

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

In three dimensions, the hyperoctahedral group is known as {{math|O × S{{sub|2}}}} where {{math|O ≅ S{{sub|4}}}} is the octahedral group, and {{math|S{{sub|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

File:Hyperoctahedral group 2; passive prefix.svg

File:3-ary Boolean functions; cube permutations; 5.svg

Hyperoctahedral groups in the n-th dimension are isomorphic to S_2 \wr S_n (\wr denots the Wreath product) and can be named as Bn, a bracket notation, or as a Coxeter group graph:

class="wikitable"

!n

!Symmetry
group

!Bn

!colspan=2|Coxeter notation

!Order

!Mirrors

!Structure

!Related regular polytopes

align=center

!2

|D4 (*4•)

|B2

[4]{{CDD|node|4|node}}222! = 84

| Dih_4 \cong S_2 \wr S_2

|Square, octagon

align=center

!3

|Oh (*432)

|B3

[4,3]{{CDD|node|4|node|3|node}}233! = 483+6

| S_4 \times S_2 \cong S_2 \wr S_3

|Cube, octahedron

align=center

!4

|±1/6[OxO].2 {{harvnb|Conway|Smith|2003}}
(O/V;O/V)* {{harvnb|du Val|1964|loc=#47}}

|B4

[4,3,3]{{CDD|node|4|node|3|node|3|node}}244! = 3844+12

| S_2 \wr S_4

|Tesseract, 16-cell, 24-cell

align=center

!5

| 

|B5

[4,3,3,3]{{CDD|node|4|node|3|node|3|node|3|node}}255! = 38405+20

| S_2 \wr S_5

|5-cube, 5-orthoplex

align=center

!6

| 

|B6

[4,34]{{CDD|node|4|node|3|node|3|node|3|node|3|node}}266! = 460806+30

| S_2 \wr S_6

|6-cube, 6-orthoplex

align=center

!...n

| 

|Bn

[4,3n-2]{{CDD|node|4|node|3|node|3}}...{{CDD|3|node|3|node}}2nn! = (2n)!!n2

| 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 \{\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.

File:Sphere symmetry group td.png in three dimensions, order 24]]

The hyperoctahedral subgroup, Dn by dimension:

class="wikitable"

!n

!Symmetry
group

!Dn

!colspan=2|Coxeter notation

!Order

!Mirrors

!Related polytopes

align=center

!2

|D2 (*2•)

|D2

[2] = [ ]×[ ]{{CDD|nodes}}42

|Rectangle

align=center

!3

|Td (*332)

|D3

[3,3]{{CDD|node|split1|nodes}}246

|tetrahedron

align=center

!4

|±1/3[Tx{{Overline|T}}].2
(T/V;T/V)−* {{harvnb|du Val|1964|loc=#42}}

|D4

[31,1,1]{{CDD|node|3|node|split1|nodes}}19212

|16-cell

align=center

!5

| 

|D5

[32,1,1]{{CDD|node|3|node|3|node|split1|nodes}}192020

|5-demicube

align=center

!6

| 

|D6

[33,1,1]{{CDD|node|3|node|3|node|3|node|split1|nodes}}2304030

|6-demicube

align=center

!...n

| 

|Dn

[3n-3,1,1]{{CDD|node|3|node}}...{{CDD|3|node|split1|nodes}}2n-1n!n(n-1)

|demihypercube

File:Sphere symmetry group th.png in three dimensions, order 24]]

File:Sphere symmetry group o.png in three dimensions, order 24]]

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

class="wikitable"

!n

!Symmetry
group

!colspan=2|Coxeter notation

!Order

align=center

!2

|C4 (4•)

[4]+{{CDD|node_h2|4|node_h2}}4
align=center

!3

|O (432)

[4,3]+{{CDD|node_h2|4|node_h2|3|node_h2}}24
align=center

!4

|1/6[O×O].2
(O/V;O/V) {{harvnb|du Val|1964|loc=#27}}

[4,3,3]+{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2}}192
align=center

!5

| 

[4,3,3,3]+{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2|3|node_h2}}1920
align=center

!6

| 

[4,3,3,3,3]+{{CDD|node_h2|4|node_h2|3|node_h2|3|node_h2|3|node_h2|3|node_h2}}23040
align=center

!...n

| 

[4,(3n-2)+]{{CDD|node_h2|4|node_h2|3|node_h2}}...{{CDD|3|node_h2|3|node_h2}}2n-1n!

Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:{{harvnb|Coxeter|1999|p=121, Essay 5 Regular skew polyhedra}} These groups have n orthogonal mirrors in n-dimensions.

class="wikitable"

!n

!Symmetry
group

!colspan=2|Coxeter notation

!Order

!Mirrors

!Related polytopes

align=center

!2

|D2 (*2•)

[4,1+]=[2]{{CDD|node|4|node_h2}}42

|Rectangle

align=center

!3

|Th (3*2)

[4,3+]{{CDD|node|4|node_h2|3|node_h2}}243

|snub octahedron

align=center

!4

|±1/3[T×T].2
(T/V;T/V)* {{harvnb|du Val|1964|loc=#41}}

[4,(3,3)+]{{CDD|node|4|node_h2|3|node_h2|3|node_h2}}1924

|snub 24-cell

align=center

!5

| 

[4,(3,3,3)+]{{CDD|node|4|node_h2|3|node_h2|3|node_h2|3|node_h2}}19205

|

align=center

!6

| 

[4,(3,3,3,3)+]{{CDD|node|4|node_h2|3|node_h2|3|node_h2|3|node_h2|3|node_h2}}230406

|

align=center

!...n

| 

[4,(3n-2)+]{{CDD|node|4|node_h2|3|node_h2}}...{{CDD|3|node_h2|3|node_h2}}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.

=H<sub>1</sub>: 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.

=H<sub>2</sub>: Schur multipliers=

The second homology groups, known classically as the Schur multipliers, were computed in {{Harv|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

{{reflist}}

References

{{refbegin}}

  • {{cite journal| 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 = 203–6| volume=24|issue=4| doi-access=free}}

  • {{cite book |author-link=Patrick du Val |first=P. |last=du Val |title=Homographies, Quaternions and Rotations |series=Oxford mathematical monographs |publisher=Clarendon Press |location= |date=1964 |oclc=904102141 |pages= |url=http://catalog.hathitrust.org/api/volumes/oclc/808822.html}}
  • {{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 | 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 | 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 (mathematician) | 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 }}
  • {{cite book |first1=H.S.M. |last1=Coxeter |first2=W.O.J. |last2=Moser |title=Generators and Relations for Discrete Groups |publisher=Springer |date=2013 |orig-date=1980 |isbn=978-3-662-21943-0 |page=92 §7.4 Linear fractional groups, p. 122 §9.3 Finite Groups|edition=4th |url={{GBurl|pGjyCAAAQBAJ|pg=PP8}}}}
  • {{cite journal|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 | issue=11 | page = 3171| bibcode=1984JMP....25.3171B }}

  • {{cite journal|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| hdl=2027.42/30235 | hdl-access=free }}

  • {{cite book |author1-link=H.S.M. Coxeter |first1=H.S.M. |last1=Coxeter |title=The Beauty of Geometry: Twelve Essays |publisher=Dover |location= |date=1999 |isbn=0-486-40919-8 |lccn=99035678 |url=}}
  • {{cite book |author-link=John Horton Conway |first1=John H. |last1=Conway |last2=Smith|first2=Derek A. |title=On Quaternions and Octonions |publisher=CRC Press |location= |date=2003 |isbn=978-1-000-68777-4 |pages= |url={{GBurl|XHKhDwAAQBAJ|pg=PT5}}}}

{{refend}}

Category:Finite reflection groups