# Geometric interpretation of exceptional symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are related to real forms of exceptional Lie algebra. Most of these symmetric spaces admit at least one geometric interpretation usually in terms of complex and real Grassmannians and their generalizations to quaternions and octonions($$\mathbb{H}$$), octonions($$\mathbb{\mathbb{O}}$$), bioctonions ($$\mathbb{C}\otimes \mathbb{O}$$), quateroctonions ($$\mathbb{H}\otimes \mathbb{O}$$) and octooctonions ($$\mathbb{O}\otimes \mathbb{O}$$). See for example the Wikipedia's entry for Symmetric Spaces.

Two of the exceptional symmetric spaces, don't seem to have such a geometric interpretation as far as I know. In Cartan notation, these two spaces are called $$EI$$ and $$EV$$ and correspond, respectively, to the exceptional symmetric spaces $$\frac{E_6}{\mathrm{Sp}(4)/ \mathbb{Z}_2}$$ and $$\frac{E_7}{\mathrm{SU}(8)/\mathbb{Z}_2 }$$ of respective ranks and dimensions $$(6,42)$$ and $$(7,70)$$.

Now that the stage is set, here is my question:

What is the geometric description of the symmetric spaces $$\frac{E_{7}}{SU(8)/ \mathbb{Z}_2}$$ and $$\frac{E_6}{Sp(4)/ \mathbb{Z}_2}$$?

References on the subject are also welcome. This question is motivated by an answer to this MO question.

In order to give a more precise idea of the kind of answer I expect, let me give some examples: the symmetric space $$\frac{F_4}{\mathrm{Spin}(9)}$$ is geometrically described as the Cayley projective plane $$\mathbb{O}P^2$$, the space $$\frac{E_6}{\mathrm{SO}(10) \mathrm{SO}(2)}$$ is geometrically the Caylay bioctonion plane $$(\mathbb{C}\otimes \mathbb{O}) \mathbb{P}^2$$ and the symmetric space $$\frac{E_6}{F_4}$$ is the space of isometrically equivalent collineations of the Cayley plane $$\mathbb{O}\mathbb{P}^2$$.

NB:These two spaces also show up as scalar manifolds in maximal supergravity theories, this is for example review in this article of Boya. But for this question, I won't consider supergravity as a geometric interpretation.

Richard Borcherds has provided an answer thanks to a reference to the book "Einstein manifolds", but the book gives the answer without any proofs or explanation. So we now have an answer but we don't understand it. So if anyone could help with explaining how "antichains" enter the story, it will be highly appreciated. I have put some extra information in the comments.

• Though I can't provide any direct answer, I wonder if you have looked at the book Spaces of Constant Curvature by J.A. Wolf or related papers? Contacting him directly at UC Berkeley might be useful, since he has gone beyond Elie Cartan in some respects and has worked for a long time with symmetric spaces. Aug 19, 2010 at 15:03
• The link to the article of Boya at springerlink.com is broken. I'm also unable to find any copy saved on the Wayback Machine. Aug 8, 2022 at 19:42

Section K of chapter 10 of Einstein manifolds by A. Besse gives geometric interpretations for all irreducible symmetric spaces. The 2 cases you ask about are apparently antichains in (H⊗Ca)P2, and (C⊗Ca)P2 (whatever this means).

• There's a wikipage on antichains, but how this relates to what's written in that table in Besse is unclear to me: en.wikipedia.org/wiki/Antichain Aug 20, 2010 at 8:50
• Here $(C\otimes Ca)P^2$ and and $(H\otimes Ca)P^2$ are the projective planes constructed respectively over the algebra of bi-octonions $\mathbb{C}\otimes \mathbb{O}$ and quateroctinions $\mathbb{H}\otimes \mathbb{O}$. Here $Ca$ is just the algebra $\mathbb{O}$ of octonions. Now $E_6\simeq \mathrm{isom}((C\otimes Ca)P^2)$ and $E_7\simeq \mathrm{isom}((H\otimes Ca)P^2)$. I think in this context "antichain" is defined with using inclusion as the order, but I am not really sure so I still don't understand the answer. I have checked the book but it does not explain the result.
– JME
Aug 22, 2010 at 17:38

Eschenburg present $E_I$ as $\{G_2\mathbb H^4/\mathbb Z_2 \subset \mathbb C \otimes \mathbb OP^2\}$ and $E_V$ as $\{G_4\mathbb C^8/\mathbb Z_2 \subset \mathbb C \otimes \mathbb OP^2\}$.

In my notes from 2008 I see interpretation of $E_I=E_6/Sp_4$ as $\{\mathbb HP^2 \times\mathbb HP^2 \subset \mathbb C \otimes \mathbb OP^2\}$ and and $E_V=E_7/SU_8$ as $\{G_2\mathbb C^6 \times G_2\mathbb C^6 \subset \mathbb C \otimes \mathbb OP^2\}$. Note that $G_2\mathbb C^6=\mathbb C \otimes \mathbb HP^2$.

Without seeing the proof we cannot tell which one is true. There are some equalities in grassmanians e.g. I believe that $G_{2,2}\mathbb C$=$G_{2,4}^+$. It is possible that for example $G_2\mathbb H^4$ is equal to $\mathbb HP^2 \times \mathbb HP^2$ or rather $S^8 \times \mathbb HP^2$ - see below.

I remember following argument for $G_{2,2}^+$. Let $1$ be fixed vector in $\mathbb R^4$. Complex structure is defined by unit vector $\color{red}v$ perpendicular to $1$. Complex lines for selected complex structure are forming $S^2=\mathbb CP^1$. Each element of $G_{2,2}^+$ belongs to exactly one complex structure, so we obtain $S^2\times S^2$.

Analogous arguments for two-dimensional grassmanian in complex and quaternion four dimensional spaces would give $\mathbb HP^1\times \mathbb CP^2$ and $\mathbb OP^1\times \mathbb HP^2$.