Plücker coordinates and the Rosenfeld planes

Abstract

The exceptional compact hermitian symmetric space EIII is the quotient $E_6/Spin\left(10\right){\times }_{Z_4}U\left(1\right)$. We introduce the Plücker coordinates which give an embedding of EIII into $C{P}^{26}$ as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.

Our motivation is to understand EIII as the complex projective octonion plane $(C\otimes O)P^2$, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety $X_{\infty }$ of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of $X_{\infty }$.

We further decompose $X=\mathrm{EIII}$ into $F_4$-orbits: $X=Y_0\cup Y_{\infty }$, where $Y_0\sim {\left(O{P}^2\right)}_C$ is an open $F_4$-orbit and is the complexification of $O{P}^2$, whereas $Y_{\infty }$ has co-dimension 1, thus EIII could be more appropriately denoted as $\overline{{\left(O{P}^2\right)}_C}$. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].