exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
The octahedral group, a finite group, is the group ofn symmetries of an octahedron.
As a symmetry group of one of the Platonic solids, the octahedral group participates in one of the three exceptional entries cases of the ADE pattern:
ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_{n \geq 1}$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ | |
D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |
More in detail, there are variants of the octahedral group corresponding to the stages of the Whitehead tower of O(3):
the full octahedral group is the subgroup of O(3)
which is the stabilizer of the standard embedding of the octahedron into Cartesian space $\mathbb{R}^3$;
the rotational octahedral group $O \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the symmetric group $S_4$;
finally the binary octahedral group is the double cover, hence the lift of $I$ to Spin(3)$\simeq$ SU(2);
next there is a string 2-group lift $String_{2O} \hookrightarrow String_{SU(2)}$ of the octahedral group (Epa 10, Epa-Ganter 16)
The group order is:
$\vert O_h\vert = 48$
$\vert O \vert = 24$
$\vert 2O \vert = 48$
The subgroup of the octahedral group on the orientation-preserving symmetries is isomorphic to the symmetric group $S_4$. This also happens to be the full tetrahedral group.
(quaternion group inside binary tetrahedral group)
The binary octahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup (normal):
In fact the only finite subgroups of SU(2) which contain $2 D_4 =Q_8$ as a proper subgroup are the exceptional ones, hence the binary tetrahedral group, the binary octahedral group and the binary icosahedral group.
See this Prop at quaternion group.
linear representation theory of binary octahedral group $2 O$
$\,$
group order: ${\vert 2O\vert} = 48$
conjugacy classes: | 1 | -1 | $i$ | a | c | e | f | g |
---|---|---|---|---|---|---|---|---|
their cardinality: | 1 | 1 | 6 | 8 | 8 | 6 | 6 | 12 |
character table over the complex numbers $\mathbb{C}$
irrep | 1 | -1 | $i$ | a | c | e | f | g |
---|---|---|---|---|---|---|---|---|
$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\rho_2$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 |
$\rho_3$ | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 |
$\rho_4$ | 3 | 3 | -1 | 0 | 0 | 1 | 1 | -1 |
$\rho_5$ | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 1 |
$\rho_6$ | 2 | -2 | 0 | 1 | -1 | $\sqrt{2}$ | $-\sqrt{2}$ | 0 |
$\rho_7$ | 2 | -2 | 0 | 1 | -1 | $-\sqrt{2}$ | $\sqrt{2}$ | 0 |
$\rho_8$ | 4 | -4 | 0 | -1 | 1 | 0 | 0 | 0 |
character table over the real numbers $\mathbb{R}$
irrep | 1 | -1 | $i$ | a | c | e | f | g |
---|---|---|---|---|---|---|---|---|
$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\rho_2$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 |
$\rho_3$ | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 |
$\rho_4$ | 3 | 3 | -1 | 0 | 0 | 1 | 1 | -1 |
$\rho_5$ | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 1 |
$\rho_6 \oplus \rho_6$ | 4 | -4 | 0 | 2 | -2 | $2 \sqrt{2}$ | $-2 \sqrt{2}$ | 0 |
$\rho_7 \oplus \rho_7$ | 4 | -4 | 0 | 2 | -2 | $-2 \sqrt{2}$ | $2 \sqrt{2}$ | 0 |
$\rho_8 \oplus \rho_8$ | 8 | -8 | 0 | -2 | 2 | 0 | 0 | 0 |
References
Groupnames, CSU(2,3)
GroupProps, Linear representation theory of binary octahedral group
Bockland, Character tables and McKay quivers (pdf)
The group cohomology of the orientation-preserving octahedral group is discussed in Groupprops, Kirdar 13.
Aspects of the linear representation theory of the binary octahedral group (irreducible representations, character table) is spelled out at
See also
Wikipedia, Octahedral symmetry
Groupprops, Group cohomology of symmetric group:S4
Mehmet Kirdar, On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters (arXiv:1309.4238)
Narthana Epa, Platonic 2-groups, 2010 (pdf)
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)
Last revised on May 15, 2019 at 05:17:40. See the history of this page for a list of all contributions to it.