Cameron-Liebler sets of generators in finite classical polar spaces

TL;DR

This paper introduces and characterizes Cameron-Liebler sets of generators in finite classical polar spaces, extending the concept from projective spaces and providing classification results.

Contribution

It generalizes Cameron-Liebler sets to finite classical polar spaces and offers multiple characterizations and classification theorems.

Findings

Characterizations of Cameron-Liebler sets in polar spaces

Classification results for these sets

Extension of the concept beyond projective spaces

Abstract

Cameron-Liebler sets were originally defined as collections of lines (`line classes') in $\mathrm{PG}(3,q)$ sharing certain properties with line classes of symmetric tactical decompositions. While there are many equivalent characterisations, these objects are defined as sets of lines whose characteristic vector lies in the image of the transpose of the point-line incidence matrix of $\mathrm{PG}(3,q)$, and so combinatorially they behave like a union of pairwise disjoint point-pencils. Recently, the concept of a Cameron-Liebler set has been generalised to several other settings. In this article we introduce Cameron-Liebler sets of generators in finite classical polar spaces. For each of the polar spaces we give a list of characterisations that mirrors those for Cameron-Liebler line sets, and also prove some classification results.