Contents

group theory

# 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 $B \iota'$ is the delooping of 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:

### 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:

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

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

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

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.

### $G$-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)$

## References

### 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)

### 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)

Last revised on April 10, 2019 at 13:54:42. See the history of this page for a list of all contributions to it.