exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
exceptional Jordan superalgebra, $K_10$
The tetrahedral group is the finite symmetry group of a tetrahedron.
As a symmetry group of one of the Platonic solids, the tetrahedral group participates in the ADE pattern:
ADE classification and McKay correspondence
More in detail, there are variants of the tetrahedral group corresponding to the stages of the Whitehead tower of O(3):
the full tetrahedral group is the subgroup of O(3)
which is the stabilizer of the standard embedding of the tetrahedron into Cartesian space $\mathbb{R}^3$;
the rotational tetrahedral group $T \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the alternating group $A_4$;
next the binary tetrahedral group $2T$ is the double cover, hence the lift of $T$ to Spin(3)$\simeq$ SU(2), this is equivalently the special linear group over the prime field $\mathbb{F}_3$
then there is a string 2-group lift $String_{2T} \hookrightarrow String_{SU(2)}$ of the tetrahedral group to a Platonic 2-group (Epa 10, Epa-Ganter 16)
The binary tetrahedral group $2T$ is, up to isomorphism, the subgroup of the group $S(\mathbb{H}) \simeq$ SU(2) $\simeq Spin(3)$ of unit quaternions consisting of the 8 unit quaternions $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by $\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$.
(These are also the vertices that span the 24-cell as a convex regular polytope in $\mathbb{R}^4$).
The full tetrahedral group is isomorphic to the symmetric group $S_4$ of permutations of four elements (see Full tetrahedral group is isomorphic to S4).
The subgroup of orientation-preserving symmetries is isomorphic to the alternating group $A_4$.
The group order is:
$\vert T_d\vert = 24$
$\vert T\vert = 12$
$\vert 2T\vert = 24$
(quaternion group inside binary tetrahedral group)
The binary tetrahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup, in fact as a normal subgroup:
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.
ADE classification and McKay correspondence
linear representation theory of binary tetrahedral group $2 T$
$\,$
group order: $\vert 2T\vert = 24$
conjugacy classes: | 1 | -1 | $i$ | a | b | c | d |
---|---|---|---|---|---|---|---|
their cardinality: | 1 | 1 | 6 | 4 | 4 | 4 | 4 |
$\,$
let $\zeta_3$ be a third root of unity, $(\zeta_3)^3 = 1$
e.g. $\zeta_3 = \tfrac{1}{2}(-1 + \sqrt{3} i)$, notice that $\zeta_3 + \zeta_3^2 = 1$
character table over the complex numbers $\mathbb{C}$
irrep | 1 | -1 | $i$ | a | b | c | d |
---|---|---|---|---|---|---|---|
$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\rho_2$ | 1 | 1 | 1 | $\zeta_3$ | $\zeta_3^2$ | $\zeta_3^2$ | $\zeta_3$ |
$\rho_2^\ast$ | 1 | 1 | 1 | $\zeta^2_3$ | $\zeta_3$ | $\zeta_3$ | $\zeta_3^2$ |
$\rho_3$ | 3 | 3 | -1 | 0 | 0 | 0 | 0 |
$\rho_4$ | 2 | -2 | 0 | $\zeta_3$ | $\zeta_3^2$ | $-\zeta_3^2$ | $-\zeta_3$ |
$\rho_4^\ast$ | 2 | -2 | 0 | $\zeta_3^2$ | $\zeta_3$ | $-\zeta_3$ | $-\zeta_3^2$ |
$\rho_5$ | 2 | -2 | 0 | 1 | 1 | -1 | -1 |
character table over the real numbers $\mathbb{R}$
irrep | 1 | -1 | $i$ | a | b | c | d |
---|---|---|---|---|---|---|---|
$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\rho_2 \oplus \rho_2^\ast$ | 2 | 2 | 2 | -1 | -1 | -1 | -1 |
$\rho_3$ | 3 | 3 | -1 | 0 | 0 | 0 | 0 |
$\rho_4 \oplus \rho_4^\ast$ | 4 | -4 | 0 | -1 | -1 | 1 | 1 |
$\rho_5 \oplus \rho_5$ | 4 | -4 | 0 | 2 | 2 | -2 | -2 |
References
The group cohomology of the tetrahedral group is discussed in Groupprops, Kirdar 13.
Discussion in the context of classification of finite rotation groups goes back to:
Exposition is in
Discussion of higher central extension to Platonic 2-groups is in
Narthana Epa, Platonic 2-groups, 2010 (pdf)
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)
See also
Mehmet Kirdar, On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters (arXiv:1309.4238)
Wikipedia, Tetrahedral symmetry
Groupprops, Group cohomology of symmetric group:S4
Last revised on May 15, 2019 at 05:05:24. See the history of this page for a list of all contributions to it.