nLab SO(3)

Idea

The special orthogonal group in dimension 3.

Properties

Cohomology

Proposition

The integral cohomology ring of the classifying space $B SO(3)$ is

$$H^\bullet\big( B SO(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3) \,,$$

where

• $p_1 \in H^4\big(B SO(3), \mathbb{Z}\big)$ is the universal first Pontryagin class;

• $W_3 = \beta(w_2) \in H^3\big( B SO(3), \mathbb{Z}\big)$ is the universal degree-3 integral Stiefel-Whitney class.

This is a special case of Brown 82, theorem 1.5, which is also reviewed as Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25.

Finite subgroups

Theorem

(ADE classification of finite subgroups of SO(3) and Spin(3)$\simeq$ SU(2))

The finite subgroups of the special orthogonal group $SO(3)$ as well as the finite subgroups of the special unitary group SU(2) are, up to conjugation, given by the following classification:

ADE classification and McKay correspondence

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 $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic 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)
D4	dihedron 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)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron 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 $SO(2n)$, $Spin(2n)$
$E_6$	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

Here under the double cover projection (using the exceptional isomorphism $SU(2) \simeq Spin(3)$)

$$SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)$$

all the finite subgroups of $SU(2)$ except the odd-order cyclic groups are the preimages of the corresponding finite subgroups of $SO(3)$, in that we have pullback diagrams

$$\array{ \left\langle \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) \right\rangle & = & \mathbb{Z}/(2n) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) } \right\rangle & = & \mathbb{Z}/n &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$$

exhibiting the even order cyclic groups as subgroups of Spin(2), including the the minimal case of the group of order 2

$$\array{ \left\langle \exp \left( \pi \mathrm{i} \right) = -1 \right\rangle & = & \mathbb{Z}/2 &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \right) } = e \right\rangle & = & 1 &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$$

as well as

$$\array{ \left\langle \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j} \right\rangle &=& 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n} \right) }, \, Ad_{\mathrm{j}} \right\rangle && D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$$

exhibiting the binary dihedral groups as sitting inside the Pin(2)-subgroup of Spin(3),

but only commuting diagrams

$$\array{ \left\langle \exp \left( 2 \pi \mathrm{i} \tfrac{1}{{2n+1}} \right) \right\rangle & = & \mathbb{Z}/(2n+1) &&\overset{\phantom{AA}}{\hookrightarrow}&& Spin(3) \\ && \big\downarrow && && \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( 2 \pi \mathrm{i} \tfrac{1}{2n+1} \right) } \right\rangle & = & \mathbb{Z}/(2n+1) &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$$

for the odd order cyclic subgroups.

This goes back to (Klein 1884, chapter I). Full proof for $SO(3)$ is spelled out for instance in (Rees 05, theorem 11, De Visscher 11). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in (Miller-Blichfeldt-Dickson 16) reviewed in (Serrano 14, section 2). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in Keenan 03, theorem 4.

Irreducible representations

$SO(3)$ has an irreducible representation as the group of rotations in $\mathbb{R}^3$, whose action preserves both the dot product and cross product. This is the “defining representation”.

But something analogous happens to be true in $\mathbb{R}^7$: There is an $SO(3)$ subgroup of the exceptional Lie group $G_2$ (see there) for which the irreducible representation of $G_2$ on $\mathbb{R}^7$ remains irreducible when restricted to this subgroup. $SO(3)$ preserves the dot and cross products defined there in terms of the imaginary octonions.

This $7$-dimensional representation may also be realized as the space of harmonic cubic homogeneous polynomials on $\mathbb{R}^3$, otherwise known as the space of $f$-orbital wavefunctions.

Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

• Euclidean group

• rigid body dynamics

References

• Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)

See also

• Wikipedia, 3D rotation group

On the integral cohomology of the classifying space:

• Edgar H. Brown, The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)

reviewed in

• Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)