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.
(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))
(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)
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. .
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:
(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:
(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:
The ordinary cohomology ring of the classifying space $B Spin(8)$ is:
1) with coefficients in the cyclic group of order 2:
where $w_i$ are the universal Stiefel-Whitney classes,
and where
2) with coefficients in the integers:
where $p_1$ is the first fractional Pontryagin class, $p_2$ is the second Pontryagin class, $\chi$ is the Euler class, and
is the Bockstein homomorphism.
Moreover, we have the following relations:
This is due to Quillen 71, Čadek-Vanžura 95, see Čadek-Vanžura 97, Lemma 4.1.
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
rotation groups in low dimensions:
sp. orth. group | spin group | pin group |
---|---|---|
SO(2) | Spin(2) | Pin(2) |
SO(3) | Spin(3) | |
SO(4) | Spin(4) | |
SO(5) | Spin(5) | |
Spin(6) | ||
Spin(7) | ||
SO(8) | Spin(8) | |
SO(9) | Spin(9) |
see also
See also
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)
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)
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