Hyperoctahedral group

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C2 group circle domains.png
The C2 group has order 8 as shown on this circle
Sphere symmetry group oh.png
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 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, ..., −1, 1, 2, ..., n } or of the set { −n, −n + 1, ..., n } such that π(i) = −π(−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. 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 (equivalently, cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry. In two dimensions, the hyperoctahedral group is known as the dihedral group of order eight, describing the symmetry of a square.

By dimension[edit]

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

n Symmetry
group
BCn Coxeter notation Order Mirrors Structure Related regular polytopes
2 D4 (*4•) BC2 [4] CDel node.pngCDel 4.pngCDel node.png 222! = 8 2 Dih_4 \cong S_2 \wr S_2 Square, octagon
3 Oh (*432) BC3 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 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]
BC4 [4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 244! = 384 4+12 S_2 \wr S_4 Tesseract, 16-cell, 24-cell
5   BC5 [4,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 255! = 3840 5+20 S_2 \wr S_5 5-cube, 5-orthoplex
6   BC6 [4,34] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 266! = 46080 6+30 S_2 \wr S_6 6-cube, 6-orthoplex
...
n   BCn [4,3n-2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png 2nn! = (2n)!! n2 S_2 \wr S_n hypercube, orthoplex

Subgroups[edit]

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.

Tetrahedral symmetry in three dimensions, order 24

The hyperoctahedral subgroup, Dn by dimension:

n Symmetry
group
Dn Coxeter notation Order Mirrors Related polytopes
2 D2 (*2•) D2 [ ]×[ ] CDel nodes.png 4 2 Rectangle
3 Td (*332) D3 [3,3] CDel node.pngCDel split1.pngCDel nodes.png 24 6 tetrahedron
4 ±1/3[TxT].2 [3]
(T/V;T/V)-* [4]
D4 [31,1,1] CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 192 12 16-cell
5   D5 [32,1,1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 1920 20 5-demicube
6   D6 [33,1,1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 23040 30 6-demicube
...n   Dn [3n-3,1,1] CDel node.pngCDel 3.pngCDel node.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 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]+ CDel node h2.pngCDel 4.pngCDel node h2.png 4
3 O (432) [4,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png 24
4 1/3[T×T].2 [5]
(T/V;T/V) [6]
[4,3,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 192
5   [4,3,3,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 1920
6   [4,3,3,3,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 23040
...n   [4,(3n-2)+] CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png...CDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 2n-1n!

Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:[7] These groups have n orthogonal mirrors in n-dimensions.

n Symmetry
group
Coxeter notation Order Mirrors Related polytopes
2 D2 (*2•) [4,1+]=[2] CDel node.pngCDel 4.pngCDel node h2.png 4 2 Rectangle
3 Th (3*2) [4,3+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png 24 3 snub octahedron
4 ±1/3[T×T].2 [8]
(T/V;T/V)* [9]
[4,(3,3)+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 192 4 snub 24-cell
5   [4,(3,3,3)+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 1920 5
6   [4,(3,3,3,3)+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 23040 6
...n   [4,(3n-2)+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png...CDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 2n-1n! n

Homology[edit]

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[edit]

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[edit]

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[edit]

  1. ^ Conway, 2003
  2. ^ Du Val, 1964
  3. ^ Conway, 2003
  4. ^ Du Val, 1964, #42
  5. ^ Conway, 2003
  6. ^ Du Val, 1964, #41
  7. ^ Coxeter (1999), p.121, Essay 5 Regular skew polyhedra
  8. ^ Conway, 2003
  9. ^ Du Val, 1964, #41

References[edit]

  • Patrick du Val, Homographies, Quaternions and Rotations (1964)
  • John Horton Conway, On Quaternions and Octonions (2003)
  • 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. 240 240, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0067943, MR 0325752 
  • Kerber, Adalbert (1975), Representations of permutation groups. II, Lecture Notes in Mathematics, Vol. 495 495, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0085740, MR 0409624 
  • Young, Alfred (1930), "On Quantitative Substitutional Analysis 5", Proceedings of the London Mathematical Society. Second Series 31: 273, doi:10.1112/plms/s2-31.1.273, ISSN 0024-6115, JFM 56.0135.02 
  • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.