Cameron-Liebler k-Classes in PG(2k+1, q)

Rodgers, Morgan; Storme, Leo; Vansweevelt, Andries

doi:10.1007/s00493-016-3482-y

Cited by 23 publications

(26 citation statements)

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“…Let f be a Boolean degree 1 function f on J q (n, k). Then f or its complement 1 − f is one of the following: 1, p + , π + , p + ∨ π + .Here p is a point, π is a hyperplane, and p / ∈ π.This improves Corollary 5.5 in [54]. In particular, we reduce the problem to the J q (n, 2) case.…”

supporting

confidence: 50%

“…For J q (n, k), some restrictions on the parameters of Boolean degree 1 functions are known, see [48,54]. Our main result for J q (n, k) is the following, which extends Theorem 1.3: Theorem 1.4.…”

mentioning

confidence: 60%

“…Let f be a Boolean degree 1 function f on J q (n, 2). Then f or its complement 1 − f is one of the following: 1, p + , π + , p + ∨ π + .Here p is a point and π is a hyperplane satisfying the condition p / ∈ π; and p + ∨ π + (S) = 1 p∈S or S⊆π .For J q (n, k), some restrictions on the parameters of Boolean degree 1 functions are known, see [48,54]. Our main result for J q (n, k) is the following, which extends Theorem 1.3: Theorem 1.4.…”

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confidence: 99%

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Boolean degree 1 functions on some classical association schemes

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Ihringer

2019

Journal of Combinatorial Theory, Series A

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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 completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

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confidence: 60%

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Boolean degree 1 functions on some classical association schemes

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Ihringer

2019

Journal of Combinatorial Theory, Series A

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“…Recently, Cameron-Liebler k-space classes (also known as Boolean degree 1 functions) received some attention [3,9,18]. In particular, Metsch showed the following [16]: Note that a tedious calculation shows that we can choose q 0 = 89 if we follow the argument in [16] without optimizing any of the used constants.…”

Section: Using a Results By Tokushige On Cross-intersecting Families Imentioning

confidence: 99%

The Chromatic Number of the $q$-Kneser Graph for Large $q$

Ihringer

2019

Electron. J. Combin.

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We obtain a new weak Hilton-Milner type result for intersecting families of k-spaces in F 2k q , which improves several known results. In particular the chromatic number of the q-Kneser graph qK n:k was previously known for n > 2k (except for n = 2k + 1 and q = 2) or k < q log q − q. Our result determines the chromatic number of qK 2k:k for q ≥ 5, so that the only remaining open cases are (n, k) = (2k, k) with q ∈ {2, 3, 4} and (n, k) = (2k + 1, k) with q = 2.

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“…We observe that all known nontrivial examples have relatively large parameter ≃ ∕ x q 2 2 , although the best known lower bound for the parameter x of a nontrivial Cameron-Liebler line class is ≳ ∕ x q 4 3 , see [20]. Secondly, for given q, one can try to verify the conjecture in n q PG( , ) for all n > 3 provided that a complete list of Cameron-Liebler line classes in q PG (3, ) is known, see ([8, Section 6.2]), [15] (see also [11,26] for a higher dimensional generalization of Cameron-Liebler line classes).…”

Section: Introductionmentioning

confidence: 99%