Contents

group theory

/clissifying

# Contents

## 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.

###### 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

###### 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.

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.

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

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

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}

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

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto 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)$

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

### 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:

### 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)