Four relations on the set of point-hyperplane anti-flags

TL;DR

This paper investigates four specific relations on the set of point-hyperplane anti-flags, revealing their recoverability properties and connections to hyperbolic polar spaces, especially over fields with two elements.

Contribution

It characterizes the interrelations among four arrangements of point-hyperplane anti-flags and links these to hyperbolic polar space structures, highlighting unique cases over binary fields.

Findings

Each relation can be derived from any other except in one case.

Over the field with two elements, one relation cannot recover the others.

A bijection exists between anti-flags and exterior points in hyperbolic polar space in this case.

Abstract

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.