The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

TL;DR

This paper explores the combinatorial structure of points and lines related to a hyperbolic quadric in PG(5,2), revealing an isomorphism with the Grassmannian G_2(8) and connections to Cayley-Dickson algebras.

Contribution

It establishes an isomorphism between a geometric configuration and the combinatorial Grassmannian G_2(8), and links this to algebraic structures like Cayley-Dickson algebras.

Findings

The configuration of points and lines is isomorphic to G_2(8).

A set of seven points corresponds to a Conwell heptad.

Removing Conwell heptads yields nested binomial configurations.

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).