Cameron–Liebler k-sets in subspaces and non-existence conditions

Abstract

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron-Liebler line classes in PG(n, q), n ≥ 3, to Cameron-Liebler sets of k-spaces in PG(n, q) and AG(n, q). In his PhD thesis, Drudge proved that every Cameron-Liebler line class in PG(n, q) intersects every 3-dimensional subspace in a Cameron-Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in PG(n, q) and AG(n, q). Together with a basic counting argument this gives a very strong non-existence condition, n ≥ 3k + 3. is condition can also be improved for k-sets in AG(n, q), with n ≥ 2k + 2.

Degree 2 Boolean Functions on Grassmann Graphs

“…Nowadays many counterexamples to the Conjecture of Cameron and Liebler are known if (n, k) = (4, 2) and q 3; see [8,14,22,23,31,33]. The general case of k > 2 has been investigated more recently, for instance see [4,15,16,26,35,41]. Indeed, it has been shown in [26] that Conjecture 5 holds for k 2 and q ∈ {2, 3, 4, 5} if we exclude the case (n, k) = (4, 2).…”

Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces

“…Over the years, there have been many interesting extensions of this result. See [2, 9, 20, 23, 29, 31] for Cameron–Liebler sets of $$k$$‐spaces in PG$$(n,q)$$, [12] for Cameron–Liebler classes in finite sets, and [19] for Cameron–Liebler sets in several classical distance‐regular graphs.…”

Equivalent definitions for (degree one) Cameron–Liebler classes of generators in finite classical polar spaces

“…The many equivalent ways to describe a Cameron-Liebler set, both algebraically and combinatorially, sparked the interest of many researchers, and allowed for generalisations. Cameron-Liebler sets of k-spaces in PG(n, q) were studied in [3,4,16,42,47], and Cameron-Liebler sets of subsets of finite sets were discussed in [19,29,43]. In [18] Cameron-Liebler sets of generators in finite classical polar spaces were introduced, and those were further investigated in [17].…”