Plücker Coordinates and the Rosenfeld Planes

Authors: Jian Qiu Preprint number: UUITP-01/24 Abstract: The exceptional compact hermitian symmetric space EIII is the quotient $E_6/SO(10)\times_{\mathbb{Z}_4}U(1)$. We introduce the Plücker coordinates which give an embedding of EIII into $\mathbb{C}P^{26}$ 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 $(\mathbb{C}\otimes\mathbb{O})P^2$, which is a piece of folklore scattered across literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety of dimension 10 denoted $X_{\infty}$. 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 $X$ into $F_4$-orbits: $X=Y_0\cup Y_{\infty}$, where $Y_0$ is an open $F_4$ orbit and is the complexified octonion projective plane and $Y_{\infty}$ has co-dimension 1, and is needed to complete $Y_0$ into a projective variety. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer \cite{Ahiezer}.