12 research outputs found
The Cameron-Liebler problem for sets
Cameron-Liebler line classes and Cameron-Liebler k-classes in PG(2k+1,q) are
currently receiving a lot of attention. Links with the Erd\H{o}s-Ko-Rado
results in finite projective spaces occurred. We introduce here in this article
the similar problem on Cameron-Liebler classes of sets, and solve this problem
completely, by making links to the classical Erd\H{o}s-Ko-Rado result on sets.
We also present a characterisation theorem for the Cameron-Liebler classes of
sets
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
Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry
In this paper we show that if is a -design of an association
scheme , and the Krein parameters vanish for
some and all (), then
consists of precisely half of the vertices of or it is
a -design, where . We then apply this result to various problems
in finite geometry. In particular, we show for the first time that nontrivial
-ovoids of generalised octagons of order do not exist. We give
short proofs of similar results for (i) partial geometries with certain order
conditions; (ii) thick generalised quadrangles of order ; (iii) the
dual polar spaces , and
, for ; (iv) the Penttila-Williford scheme. In
the process of (iv), we also consider a natural generalisation of the
Penttila-Williford scheme in , .Comment: This paper builds on part of the doctoral work of the second author
under the supervision of the first. The second author acknowledges the
support of an Australian Government Research Training Program Scholarship and
Australian Research Council Discovery Project DP20010195
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)
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 sets for maximal totally isotropic flats in classical affine spaces
Let be the -dimensional classical affine space
with parameter over a -element finite field , and be the set of all maximal totally isotropic flats in
. In this paper, we discuss Cameron-Liebler sets in
, obtain several equivalent definitions and present some
classification results.Comment: 25 page
Cameron-Liebler sets in permutation groups
Consider a group acting on a set , the vector is a
vector with the entries indexed by the elements of , and the -entry is 1
if maps to , and zero otherwise. A -Cameron-Liebler set
is a subset of , whose indicator function is a linear combination of
elements in . We investigate Cameron-Liebler
sets in permutation groups, with a focus on constructions of Cameron-Liebler
sets for 2-transitive groups.Comment: 25 page
Boolean degree 1 functions on some classical association schemes
We investigate Boolean degree 1 functions for several classical association
schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces,
and bilinear forms graphs, as well as some other domains such as multislices
(Young subgroups of the symmetric group). In some settings, Boolean degree 1
functions are also known as \textit{completely regular strength 0 codes of
covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight
sets}.
We classify all Boolean degree functions on the multislice. On the
Grassmann scheme we show that all Boolean degree functions are
trivial for , and , and that
for general , the problem can be reduced to classifying all Boolean degree
functions on . We also consider polar spaces and the bilinear
forms graphs, giving evidence that all Boolean degree functions are trivial
for appropriate choices of the parameters.Comment: 22 pages; accepted by JCTA; corrected Conjecture 6.