Small maximal partial spreads in classical finite polar spaces

Klein, Andreas; Metsch, Klaus; Storme, Leo

doi:10.1515/advgeom.2010.007

Cited by 15 publications

(7 citation statements)

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“…The number of planes through a point P / ∈ π ∪ π ′ and disjoint to π and π ′ is the number of lines in P ⊥ , disjoint to the lines corresponding to π and π ′ . By [18,Corollary 19] this number equals q 2 (q − 1), and we find:…”

Section: Classification Resultsmentioning

confidence: 88%

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

Boeck

D'haeseleer

2020

Discrete Mathematics

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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).

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Section: Classification Resultsmentioning

confidence: 88%

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

Boeck

D'haeseleer

2020

Discrete Mathematics

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“…In Table 2, we summarize the known lower bounds on the size of small maximal partial spreads of polar spaces. The results for Q + (2n + 1, q), W(2n + 1, q) and H(2n + 1, q 2 ) are proved in [7].…”

Section: Remarksmentioning

confidence: 84%

Sets of generators blocking all generators in finite classical polar spaces

Beule

Hallez

Metsch³

et al. 2013

Journal of Combinatorial Theory, Series A

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We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n, q), Q − (2n + 1, q) and H(2n, q 2 ), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.keywords: partial spreads, blocking sets, finite classical polar spaces. MSC 2010: 51E20, 51E21. Affiliations:

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“…Finally, we would like to remark that the property from Corollary 1 does not characterize the regular near 2d-gons of order (s, t) meeting the bound on t. The dual polar graph arising from the polar space W (2d − 1, q), which is of order (s, t) = (q, (q d − 1)/(q − 1) − 1), provides a counterexample if d is odd, as was worked out in [25,Theorem 21], although t 2 = s = s 2 in this case. We again state the result in an adapted form.…”

Section: Lemma 2 Let a B And C Be Maximals In The Polar Spacementioning

confidence: 96%

A Higman inequality for regular near polygons

Vanhove

2011

J Algebr Comb

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The inequality of Higman for generalized quadrangles of order (s, t) with s > 1 states that t ≤ s 2 . We generalize this by proving that the intersection number c i of a regular near 2d-gon of order (s, t) with s > 1 satisfies the tight bound c i ≤ (s 2i − 1)/(s 2 − 1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d, b, α, β) = (d, −q, −(q + 1)/2, −((−q) d + 1)/2) with q an odd prime power.We refer the reader to Sect. 2 for the definitions of, for instance, (finite) generalized polygons, near polygons, and polar spaces.Feit and Higman [18] showed that (finite) generalized n-gons of order (s, t) = (1, 1) with n ≥ 3 can only exist if n ∈ {3, 4, 6, 8, 12}; if n = 12, then s = 1 or t = 1. If s > 1, then the following inequalities must hold: if n = 4, then t ≤ s 2 [20], if n = 6, then t ≤ s 3 [19], and if n = 8, then t ≤ s 2 [21]. Bose and Shrikhande [5] also proved that if n = 4 and t = s 2 , then for any triple of nonadjacent vertices, the number

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Small maximal partial spreads in classical finite polar spaces

Cited by 15 publications

References 10 publications

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

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

Sets of generators blocking all generators in finite classical polar spaces

A Higman inequality for regular near polygons

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