Plücker coordinates and the Rosenfeld planes

The exceptional compact hermitian symmetric space EIII is the quotient E-6/Spin(10)x(Z4)U(1). We introduce the Plucker coordinates which give an embedding of EIII into CP26 as a projective subvariety. The subvariety is cut out by 27 Plucker 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 circle times 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-infinity 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-infinity. We further decompose X = EIII into F-4-orbits: X = Y-0 boolean OR Y-infinity, where Y-0 similar to (OP2)(C) is an open F-4-orbit and is the complexification of OP2, whereas Y-infinity has co-dimension 1, thus EIII could be more appropriately denoted as <((OP2)(C))over bar>. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2]. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).