The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Abstract

Given a hyperbolic quadric of PG(5,2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the $(28_6, 56_3)$-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type $G_2(8)$. It is also pointed out that a set of seven points of $G_2(8)$ whose labels share a mark corresponds to a Conwell heptad of PG(5,2). Gradual removal of Conwell heptads from the $(28_6, 56_3)$-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).