The Veldkamp Space of the Smallest Slim Dense Near Hexagon

TL;DR

This paper provides a comprehensive classification of the Veldkamp space of the smallest slim dense near hexagon, revealing its structure as isomorphic to PG(7, 2) and detailing the types and properties of its hyperplanes and lines.

Contribution

It offers the first detailed description and classification of the Veldkamp space of this near hexagon, including hyperplanes, lines, and their automorphism groups.

Findings

Veldkamp space is isomorphic to PG(7, 2)

255 hyperplanes classified into five types

10795 Veldkamp lines classified into 41 types

Abstract

We give a detailed description of the Veldkamp space of the smallest slim dense near hexagon. This space is isomorphic to PG(7, 2) and its 2^8 - 1 = 255 Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as by a sequence of the cardinalities of points of given orders and/or that of (grid-)quads of given types. For each type we also give its weight, stabilizer group within the full automorphism group of the near hexagon and the total number of copies. The totality of (255 choose 2)/3 = 10795 Veldkamp lines split into 41 different types. We give a complete classification of them in terms of the properties of their cores (i. e., subconfigurations of points and lines common to all the three hyperplanes comprising a given Veldkamp line) and the types of the hyperplanes they are composed of. These findings may lend themselves into important physical applications, especially in view of recent emergence of a variety of closely related finite geometrical concepts linking quantum information with black holes.