# The isotropy subgroup of the action of Spin(7) on the Grassmannian G(3,8)

The group Spin$$\left( 7\right) \$$is the universal cover of $$SO\left( 7\right)$$ and is characterized as the subgroup of $$SO\left( 8\right)$$ consisting of the automorphisms of the triple cross product $$X:\mathbb{R}% ^{8}\times \mathbb{R}^{8}\times \mathbb{R}^{8}\rightarrow \mathbb{R}^{8}$$ (see for instance [SW]). In particular, Spin$$\left( 7\right)$$ acts canonically on the Grassmannian $$G\left( 3,8\right)$$ of oriented 3-dimensional subspaces of $$\mathbb{R}^{8}=\mathbb{O}$$. We would like to know the isotropy subgroup $$K$$ of this action at $$1\wedge i\wedge j$$.

What we know:

Theorem 9.3 in [SW] asserts that Spin$$\left( 7\right)$$ acts transitively on $$\begin{equation*} \{(u,v,w,x)\in \mathbb{R}^{8}\mid u,v,w,X(u,v,w),x\text{ are orthonormal}\}% \text{.} \end{equation*}$$ This implies the following:

• The action of Spin$$\left( 7\right)$$ on $$G\left( 3,8\right)$$ is transitive and so the dimension of $$K$$ must be $$6$$.

• Given a positively oriented orthonormal basis $$w_{1},w_{2},w_{3}$$ of $$% 1\wedge i\wedge j$$, then there exist $$g\in$$ Spin$$~\left( 7\right)$$ satisfying $$g\left( 1\right) =w_{1}$$, $$g\left( i\right) =w_{2}$$ and $$g\left( j\right) =w_{3}$$ (we recall that $$X\left( 1,i,j\right) =k$$).

On the other hand, we know that the group $$S^{3}\times S^{3}\subset G_{2}\subset$$ Spin$$\left( 7\right)$$ acts on the octonions as follows: $$\begin{equation*} \left( u,v\right) \cdot \left( x,y\right) =\left( ux\bar{u},vy\bar{u}\right) \text{,} \end{equation*}$$ where $$\left( x,y\right) \in \mathbb{H}\times \mathbb{H}=\mathbb{O}$$ (as sets). This gives us a 4-dimensional subgroup of $$K$$ given by $$\begin{equation*} \left\{ \left( e^{tk},v\right) \mid t\in \mathbb{R},v\in S^{3}\right\}, \end{equation*}$$ since conjugation by $$e^{tk}$$ fixes 1 and rotates the oriented plane $$i\wedge j$$ through the angle $$2t$$. In particular, if $$w_{1}=1$$, $$w_{2}=\cos t~i+\sin t~j$$ and $$w_{3}=\cos t~j-\sin t~i$$, then $$g$$ as above can be taken to be $$g\left( x,y\right) =\left( e^{tk/2},1\right) \cdot \left( x,y\right)$$ .

We would prefer to have the elements of $$K$$ explicit as linear isometries of $$\mathbb{R}^{8},$$ but, of course, it would be interesting to us to know the Lie algebra of $$K$$ as a subset of $$o(8)$$ or the isomorphism type of $$K$$.

[SW] D.A. Salamon, T. Walpuski, Notes on the octonions. Proceedings of the Gökova Geometry-Topology Conference 2016, 1--85, Gökova Geometry/Topology Conference (GGT), Gökova, 2017. Also: arXiv:1005.2820 [math.RA]