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 E6/SO(10)×Z4 U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP26 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⊗O)P2, 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∞. 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∞. We further decompose X into F4-orbits: X=Y0∪Y∞, where Y0 is an open F4 orbit and is the complexified octonion projective plane and Y∞ has co-dimension 1, and is needed to complete Y∞ into a projective variety. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer.