A combinatorial characterisation of embedded polar spaces

TL;DR

This paper provides a combinatorial characterization of embedded polar spaces, showing that certain generator sets correspond precisely to embedded polar spaces within larger polar spaces.

Contribution

It introduces a new combinatorial criterion to identify sets of generators that form embedded polar spaces, advancing the understanding of their structure.

Findings

Sets of generators with specific combinatorial properties are exactly those of embedded polar spaces.

The characterization helps distinguish embedded polar spaces from other configurations.

Provides a framework for identifying embedded polar spaces in larger polar spaces.

Abstract

Some classical polar spaces admit polar spaces of the same rank as embedded polar spaces (often arisen as the intersection of the polar space with a non-tangent hyperplane). In this article we look at sets of generators that behave combinatorially as the set of generators of such an embedded polar space, and we prove that they are the set of generators of an embedded polar space.