Cameron-Liebler sets of generators in finite classical polar spaces

Maarten De Boeck, Morgan Rodgers, Leo Storme, Andrea Svob

Published 2017-12-17Version 1

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 disjoint union of point-pencils. Recently, the concept of a Cameron-Liebler set has been generalized 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 except $\mathcal{W}(2n+1,q)$, $n\geq3$ and $q$ odd, we give a list of characterisations that mirrors those for Cameron-Liebler line sets, and also prove some classification results.