On collineations and dualities of finite generalized polygons

Temmermans, Beukje; Thas, J. A.; Maldeghem, Hendrik Van

doi:10.1007/s00493-009-2435-0

Cited by 11 publications

(20 citation statements)

References 13 publications

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“…By extracting the main ideas and fine-tuning them, we are able to give a self-contained proof, which in the end even leads to a somewhat stronger result. We note that similar more general techniques and results on generalized hexagons (but not our main results) have also been obtained by Temmermans, Thas, and Van Maldeghem [27].…”

Section: Generalized Polygonssupporting

confidence: 86%

Distance-regular Cayley graphs with small valency

Dam

Jazaeri

2019

Ars Math. Contemp.

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We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most 4, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 5, and the Cayley graphs among all distance-regular graphs with girth 3 and valency 6 or 7. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some "exceptional" distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.

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Section: Generalized Polygonssupporting

confidence: 86%

Distance-regular Cayley graphs with small valency

Dam

Jazaeri

2019

Ars Math. Contemp.

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“…In light of Corollary 1.14 it is natural that the theory of domesticity in rank 2 buildings (ie, generalised polygons) plays a central role. This analysis has been undertaken in [9,14,17], and since only residues of types A 1 × A 1 , A 2 , and B 2 /C 2 appear as rank 2 residues of irreducible thick spherical buildings of rank 3 or more, the relevant results are as follows.…”

Section: Background and Definitionsmentioning

confidence: 99%

Opposition diagrams for automorphisms of large spherical buildings

Parkinson¹,

Maldeghem²

2019

Journal of Combinatorial Theory, Series A

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Let θ be an automorphism of a thick irreducible spherical building ∆ of rank at least 3 with no Fano plane residues. We prove that if there exist both type J 1 and J 2 simplices of ∆ mapped onto opposite simplices by θ, then there exists a type J 1 ∪ J 2 simplex of ∆ mapped onto an opposite simplex by θ. This property is called cappedness. We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano plane residues. In these buildings automorphisms are not necessarily capped.

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“…No duality of a thick generalised 2n-gon is anisotropic. Moreover, if Γ is a finite thick generalised 2n-gon with parameters (s, t) then: s and t are coprime (see also [23,Corollary 5.2]). Suppose further that |L 4 | = 0 (and so θ is anisotropic).…”

Section: Anisotropic Automorphismsmentioning

confidence: 99%

The Combinatorics of Automorphisms and Opposition in Generalised Polygons

2015

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We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters.

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On collineations and dualities of finite generalized polygons

Cited by 11 publications

References 13 publications

Distance-regular Cayley graphs with small valency

Distance-regular Cayley graphs with small valency

Opposition diagrams for automorphisms of large spherical buildings

The Combinatorics of Automorphisms and Opposition in Generalised Polygons

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