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exceptional Jordan superalgebra, $K_10$
The octahedral group, a finite group, is the group of 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
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 $S_4$;
the binary octahedral group is the double cover, hence the lift of $O$ 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
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, Tomoda & Zvengrowski 08, Sec. 4.2, Kirdar 13, Epa & Ganter 16, p. 12.
Aspects of the linear representation theory of the binary octahedral group (irreducible representations, character table) is spelled out at
See also
Discussion of the group cohomology:
Groupprops, Group cohomology of symmetric group:S4
Satoshi Tomoda, Peter Zvengrowski, Remarks on the cohomology of finite fundamental groups of 3-manifolds, Geom. Topol. Monogr. 14 (2008) 519-556 (arXiv:0904.1876)
Mehmet Kirdar, On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters [arXiv:1309.4238]
Discussion of Platonic 2-group-extensions:
Narthana Epa, Platonic 2-groups, 2010 (pdf)
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)
Last revised on June 23, 2023 at 10:39:14. See the history of this page for a list of all contributions to it.