Contents

group theory

# Contents

## Idea

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:

Dynkin diagram/
Dynkin quiver
dihedron,
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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic group of order 3
$\mathbb{Z}_3$
cyclic group of order 3
$\mathbb{Z}_3$
SU(3)
A3
=
D3
cyclic group of order 4
$\mathbb{Z}_4$
cyclic group of order 4
$2 D_2 \simeq \mathbb{Z}_4$
SU(4)
$\simeq$
Spin(6)
D4dihedron on
bigon
Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
$D_8$
binary dihedral group of order 16
$2 D_{8}$
SO(12), Spin(12)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group, spin group
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral 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 tetrahedral group corresponding to the stages of the Whitehead tower of O(3):

• the full tetrahedral group is the subgroup of O(3)

$T_d \hookrightarrow O(3)$

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 $\mathbb{F}_3$

$2T \simeq SL(2,\mathbb{F}_3)$
• 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)

$\array{ String_{2T} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 T &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ T \simeq A_4 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ T_d \simeq S_5 &\hookrightarrow & O(3) }$

## Definition

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 $\mathbb{R}^4$).

## Properties

### Basic properties

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:

$\vert T_d\vert = 24$

$\vert T\vert = 12$

$\vert 2T\vert = 24$

###### Proposition

(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:

$2 D_4 =Q_8 \subset 2 T \,.$

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.

### As part of the ADE pattern

Dynkin diagram/
Dynkin quiver
dihedron,
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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic group of order 3
$\mathbb{Z}_3$
cyclic group of order 3
$\mathbb{Z}_3$
SU(3)
A3
=
D3
cyclic group of order 4
$\mathbb{Z}_4$
cyclic group of order 4
$2 D_2 \simeq \mathbb{Z}_4$
SU(4)
$\simeq$
Spin(6)
D4dihedron on
bigon
Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
$D_8$
binary dihedral group of order 16
$2 D_{8}$
SO(12), Spin(12)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group, spin group
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral 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

### Character table

linear representation theory of binary tetrahedral group $2 T$

$\,$

group order: $\vert 2T\vert = 24$

conjugacy classes:1-1$i$abcd
their cardinality:1164444

$\,$

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 $\mathbb{C}$

irrep1-1$i$abcd
$\rho_1$1111111
$\rho_2$111$\zeta_3$$\zeta_3^2$$\zeta_3^2$$\zeta_3$
$\rho_2^\ast$111$\zeta^2_3$$\zeta_3$$\zeta_3$$\zeta_3^2$
$\rho_3$33-10000
$\rho_4$2-20$\zeta_3$$\zeta_3^2$$-\zeta_3^2$$-\zeta_3$
$\rho_4^\ast$2-20$\zeta_3^2$$\zeta_3$$-\zeta_3$$-\zeta_3^2$
$\rho_5$2-2011-1-1

character table over the real numbers $\mathbb{R}$

irrep1-1$i$abcd
$\rho_1$1111111
$\rho_2 \oplus \rho_2^\ast$222-1-1-1-1
$\rho_3$33-10000
$\rho_4 \oplus \rho_4^\ast$4-40-1-111
$\rho_5 \oplus \rho_5$4-4022-2-2

References

• Bockland, Character tables and McKay quivers (pdf)

### Group cohomology

The group cohomology of the tetrahedral group is discussed in Groupprops, Tomoda & Zvengrowski 08, Sec. 4.1 Kirdar 13, Epa & Ganter 16, p. 12.

Discussion in the context of classification of finite rotation groups goes back to:

• Felix Klein, chapter I.4 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

Exposition: