30 research outputs found
Cameron-Liebler sets of k-spaces in PG(n,q)
Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F.
Ihringer. We list several equivalent definitions for these Cameron-Liebler
sets, by making a generalization of known results about Cameron-Liebler line
sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also
present a classification result
Equivalent definitions for (degree one) Cameron-Liebler classes of generators in finite classical polar spaces
In this article, we study degree one Cameron-Liebler sets of generators in
all finite classical polar spaces, which is a particular type of a
Cameron-Liebler set of generators in this polar space, [9]. These degree one
Cameron-Liebler sets are defined similar to the Boolean degree one functions,
[15]. We summarize the equivalent definitions for these sets and give a
classification result for the degree one Cameron-Liebler sets in the polar
spaces W(5,q) and Q(6,q)
Translation hyperovals and F-2-linear sets of pseudoregulus type
In this paper, we study translation hyperovals in PG(2, q(k)). The main result of this paper characterises the point sets defined by translation hyperovals in the Andre/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2, q(k)) are precisely those that have a scattered F-2 -linear set of pseudoregulus type in PG(2k -1, q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2, q(2))
Regular ovoids and Cameron-Liebler sets of generators in polar spaces
Cameron-Liebler sets of generators in polar spaces were introduced a few
years ago as natural generalisations of the Cameron-Liebler sets of subspaces
in projective spaces. In this article we present the first two constructions of
non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular
m-ovoids of k-spaces are introduced as a generalization of m-ovoids of polar
spaces. They are used in one of the aforementioned constructions of
Cameron-Liebler sets
Cameron-Liebler sets of k-spaces in PG(n,q)
Cameron-Liebler sets of k-spaces were introduced recently in Filmus and Ihringer (J Combin Theory Ser A, 2019). We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n,q) and Cameron-Liebler sets of k-spaces in PG(2k+1,q). We also present some classification results
Cameron-Liebler line classes in AG(3,q)
The study of Cameron-Liebler line classes in PG() arose from classifying
specific collineation subgroups of PG(). Recently, these line classes were
considered in new settings. In this point of view, we will generalize the
concept of Cameron-Liebler line classes to AG(). In this article we define
Cameron-Liebler line classes using the constant intersection property towards
line spreads. The interesting fact about this generalization is the link these
line classes have with Cameron-Liebler line classes in PG(). Next to
giving this link, we will also give some equivalent ways to consider
Cameron-Liebler line classes in AG(), some classification results and an
example based on the example found in [3] and [6]