Plücker Coordinates and the Rosenfeld Planes

Abstract

The exceptional compact hermitian symmetric space EIII is the quotient $E_6/Spin(10){\times }_{{\mathbb{Z}}_4}U(1)$. 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 $(\mathbb{C}\otimes \mathbb{O})P^2$, 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 $Y_0\sim (\mathbb{O}{P}^2{)}_{\mathbb{C}}$ is an open $F_4$-orbit and is the complexification of $\mathbb{O}{P}^2$, whereas $Y_{\infty }$ has co-dimension 1, thus EIII could be more appropriately denoted as $\overline{(\mathbb{O}{P}^2{)}_{\mathbb{C}}}$. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer \cite{Ahiezer}.

44 pages, final version published in J.Geom.Phys