nLab SO(8)

Idea

Among all special orthogonal groups $SO(n)$, the case of $SO(8)$ is special, since in the ADE classification of simple Lie groups it corresponds to D4, which makes its representation theory enjoy triality.

Properties

Subgroup lattice

Proposition

(Spin(7)-subgroups in Spin(8))

There are precisely 3 conjugacy classes of Spin(7)-subgroups inside Spin(8), and the triality group $Out(Spin(8))$ acts transitively on these three classes.

(Varadarajan 01, Theorem 5 on p. 6, see also Kollross 02, Prop. 3.3 (1))

Proposition

(G₂ is intersection of Spin(7)-subgroups of Spin(8))

The intersection inside Spin(8) of any two Spin(7)-subgroups from distinct conjugacy classes of subgroups (according to Prop. ) is the exceptional Lie group G₂, hence we have pullback squares of the form

(Varadarajan 01, Theorem 5 on p. 13)

Proposition

We have the following commuting diagram of subgroup inclusions, where each square exhibits a pullback/fiber product, hence an intersection of subgroups:

Here in the bottom row we have the Lie groups

Spin(5)$\hookrightarrow$ Spin(6) $\hookrightarrow$ Spin(7) $\hookrightarrow$ Spin(8)

with their canonical subgroup-inclusions, while in the top row we have

SU(2)$\hookrightarrow$ SU(3) $\hookrightarrow$ G₂ $\hookrightarrow$ Spin(7)

and the right vertical inclusion $\iota'$ is one of the two non-standard inclusions, according to Prop. .

Proof

The square on the right is that from Prop. .

The square in the middle is Varadarajan 01, Lemma 9 on p. 10.

The statement also follows with Onishchik 93, Table 2, p. 144:

Proposition

(Spin(5).Spin(3)-subgroups in SO(8))

The direct product group SO(3) $\times$ SO(5) together with the groups Sp(2).Sp(1) and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into SO(8), form 3 conjugacy classes of subgroups inside SO(8), and the triality group $Out(Spin(8))$ acts transitively on these three classes.

(Kollross 02, Prop. 3.3 (3))

Similarly:

Proposition

(Spin(5).Spin(3)-subgroups in Spin(8))

The groups Spin(5).Spin(3), Sp(2).Sp(1) and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into Spin(8), form 3 conjugacy classes of subgroups inside Spin(8), and the triality group $Out(Spin(8))$ acts transitively on these three classes.

(Čadek-Vanžura 97, Sec. 2)

In summary we have these subgroup inclusions

permuted by triality:

graphics grabbed from FSS 19, Sec. 3.3

Homotopy groups

The homotopy groups of $SO(8)$ in low degrees:

$G$	$\pi_1$	$\pi_2$	$\pi_3$	$\pi_4$	$\pi_5$	$\pi_6$	$\pi_7$	$\pi_8$	$\pi_9$	$\pi_10$	$\pi_11$	$\pi_12$	$\pi_13$	$\pi_14$	$\pi_15$
$SO(8)$	$\mathbb{Z}_2$	$0$	$\mathbb{Z}$	$0$	$0$	$0$	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 3}$	$\mathbb{Z}_{2}^{\oplus 3}$	$\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$	$\mathbb{Z}_2 \oplus \mathbb{Z}$	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$	$\mathbb{Z}_2^{\oplus 7}$

Cohomology of classifying spaces

Proposition

The ordinary cohomology ring of the classifying space $B Spin(8)$ is:

1) with coefficients in the cyclic group of order 2:

$$H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \;\simeq\; \mathbb{Z} \big[ w_4, w_6, w_7, w_8, \; \rho_2 \left( \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) \right) \big]$$

where $w_i$ are the universal Stiefel-Whitney classes,

and where

$$\rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)$$

is mod 2 reduction

2) with coefficients in the integers:

$$H^\bullet \big( B Spin(8), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \Big[ \tfrac{1}{2}p_1, \; \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \right) - \tfrac{1}{2}\chi , \; \chi, \; \beta(w_6) \Big] / \big\langle 2 \beta(w_6)\big\rangle \,,$$

where $p_1$ is the first fractional Pontryagin class, $p_2$ is the second Pontryagin class, $\chi$ is the Euler class, and

$$\beta \;\colon\; H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \longrightarrow H^{\bullet + 1} \big( B Spin(8), \mathbb{Z} \big)$$

is the Bockstein homomorphism.

Moreover, we have the following relations:

$$\begin{aligned} \rho_2\left( \tfrac{1}{2}p_1 \right) & = w_4 \\ \rho_2\big( \chi\big) & = w_8 \end{aligned}$$

This is due to Quillen 71, Čadek-Vanžura 95, see Čadek-Vanžura 97, Lemma 4.1.

Proposition

Consider the delooping of the triality automorphism relating Sp(2).Sp(1) with Spin(5).Spin(3) (Prop. ) on classifying spaces

$$\array{ B \big( Spin(5) \cdot Spin(3) \big) &\hookrightarrow& B Spin(8) \\ \big\downarrow && \big\downarrow^{ B \mathrlap{tri} } \\ B \big( Sp(2) \cdot Sp(1) \big) &\hookrightarrow& B Spin(8) }$$

Then the pullback of the universal characteristic classes of $B Spin(8)$ (from Prop. ) along $B tri$ is as follows:

$$\big( B tri \big)^\ast \;\colon\; \begin{aligned} \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1 \\ \chi & \mapsto - \tfrac{1}{4} \big( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \big) + \tfrac{1}{2}\chi \\ \tfrac{1}{4} \big( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \big) - \tfrac{1}{2}\chi & \mapsto - \chi \end{aligned}$$

(Čadek-Vanžura 97, Lemma 4.2)

In fact $tri^{-1} = tri$.

Hence, in rational cohomology:

$$\begin{aligned} \big( B tri \big)^\ast \big( \tfrac{1}{4}p_2 \big) & = \big( B tri \big)^\ast \Big( \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) + \big(\tfrac{1}{4}p_1\big)^2 + \tfrac{1}{2}\chi \Big) \\ & = -\chi + \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2} \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) \end{aligned}$$

$G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

Octonionic construction of representations

The group $Spin(8)$ has three 8-dimensional irreducible real representations: the right- and left-handed spinor representations $S_+$ and $S_-$, and the vector representation $V$, which factors through $SO(8)$. (Bryant 20) gives a unified construction of all three representations using octonions, as follows.

For $x \in \mathbb{O}$ define a linear map $m_x \colon \mathbb{O}^2 \to \mathbb{O}^2$ as follows:

$$m_x = \left( \begin{array}{cc} 0 & C R_x \\ -C L_x & 0 \end{array} \right)$$

where $L_x, R_x, C \colon \mathbb{O} \to \mathbb{O}$ are left multiplication by $x$, right multiplication by $x$ and octonionic conjugation, respectively. Since $m_x^2 = -{|x|}^2$ this map induces an action of the Clifford algebra $Cliff(\mathbb{O})$ on $\mathbb{O}^2$. (To be precise, this is Clifford algebra generated by 8 anticommuting square roots of minus 1). One can show that this action gives an isomorphism of real associative algebras

$$Cliff(\mathbb{O}) \cong End(\mathbb{O}^2)$$

where $End(\mathbb{O}^2)$, the algebra of all real-linear transformations of $\mathbb{O}^2$, is isomorphic to the algebra of $16 \times 16$ real matrices.

The group $Spin(8)$ is isomorphic to the group of linear transformations generated by products $m_x m_y$ where $x, y \in \mathbb{O}$ have ${|x|} = {|y|} = 1$. This group is also generated by elements

$$\left( \begin{array}{cc} L_u & 0 \\ 0 & R_u \end{array} \right)$$

where $u \in \mathbb{O}$ has ${|u|} = 1$. In this sense we may say $Spin(8)$ is generated by unit octonions $u$, though octonion multiplication is not itself associative.

It follows that $\mathbb{O}^2$ splits, as a representation of $Spin(8)$, into two summands isomorphic to $\mathbb{O}$. These are the two spinor representations of $Spin(8)$. To match standard conventions Bryant calls the first summand, on which $u$ acts as $L_u$, the spinor representation $S_-$, and the second, on which $u$ acts as $R_u$, the spinor representation $S_+$. The vector representation of $Spin(8)$ can also be identified with $\mathbb{O}$, with $u$ acting as $L_u R_u$.

Bryant also describes the intertwiners $V \otimes S_{\pm} \to S_{\mp}$ and the triality automorphisms of $Spin(8)$ in these terms.

Related concepts

• principal SO(8)-bundle

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

References

General

• Robert Bryant, Notes on spinors in low dimension. (arXiv:2011.05568)

See also

• Wikipedia, SO(8)

Subgroup lattice

On the subgroup lattice of Spin(8)

• A. L. Onishchik (ed.) Lie Groups and Lie Algebras

  • I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

  • II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

• Veeravalli Varadarajan, Spin(7)-subgroups of SO(8) and Spin(8), Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, pdf)

• Martin Čadek, Jiří Vanžura, On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

• Megan M. Kerr, New examples of homogeneous Einstein metrics, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (euclid:1030132086)

• Andreas Kollross, Prop. 3.3 of A Classification of Hyperpolar and Cohomogeneity One Actions, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)

Discussion with an eye towards foundations of M-theory:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M-theory anomaly cancellation (arXiv:1904.10207)

Cohomology

The integral cohomology of the classifying spaces $B SO(8)$ and $B Spin(8)$ and the action of triality on these is discussed in

• Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)

• Daniel Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann . 194 (1971), 19

• Martin Čadek, Jiří Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 (dml-cz:118764)

• Martin Čadek, Jiří Vanžura, Section 2 of On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)