Domesticity in Generalized Quadrangles

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

doi:10.1007/s00026-012-0145-6

Cited by 9 publications

(30 citation statements)

References 11 publications

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“…For rank 2, there are counterexamples in the case of finite generalized quadrangles (with non-classical parameters, see [11]). …”

Section: Lemma 53 the Duality θ Is Line-domesticmentioning

confidence: 99%

“…In the spherical case, they proved that only the identity maps no flag to an opposite flag. This led Temmermans, Thas and the present author [10,11,12] to a more refined definition of J-domestic automorphism: this is an automorphism of a spherical building over the type set I ⊇ J mapping no flag of type J to an opposite flag. For J = I, we obtain domesticity as defined above.…”

Section: Introductionmentioning

confidence: 99%

“…Hence in order to understand all J-domestic automorphisms, one has to classify the domestic automorphisms. This was done for projective spaces in [10] and for generalized quadrangles in [11]. Partial-but important as we shall see in the proof of Lemma 5.3-results for polar spaces are contained in [12].…”

Section: Introductionmentioning

confidence: 99%

“…Such automorphisms of generalized quadrangles are classified in [11]; there are exactly three conjugacy classes of these, they all have order 4 and exist in the small generalized quadrangles of orders (2,2), (2,4) and (3,5). It is an open question whether domestic automorphisms admitting no domesticity diagram exist in spherical buildings of rank at least 3.…”

Section: Introductionmentioning

confidence: 99%

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Symplectic polarities of buildings of type E 6

Maldeghem

2011

Des. Codes Cryptogr.

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A symplectic polarity of a building ∆ of type E 6 is a polarity whose fixed point structure is a building of type F 4 containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality θ of ∆ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E 6 of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E 6 for which the Phan geometry is empty.

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“…For rank 2, there are counterexamples in the case of finite generalized quadrangles (with non-classical parameters, see [11]). …”

Section: Lemma 53 the Duality θ Is Line-domesticmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symplectic polarities of buildings of type E 6

Maldeghem

2011

Des. Codes Cryptogr.

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“…But this is false in general, as [2] contains some counterexamples. However, we have taken a closer look at these counterexamples in [7,8] and could classify the automorphisms of projective spaces and generalized quadrangles which map no chamber to an opposite one. The bounds d then follow easily.…”

Section: Introductionmentioning

confidence: 97%

Collineations of polar spaces with restricted displacements

Temmermans

Thas

Maldeghem

2011

Des. Codes Cryptogr.

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Let J be a set of types of subspaces of a polar space. A collineation (which is a type-preserving automorphism) of a polar space is called J -domestic if it maps no flag of type J to an opposite one. In this paper we investigate certain J -domestic collineations of polar spaces. We describe in detail the fixed point structures of collineations that are i-domestic and at the same time (i + 1)-domestic, for all suitable types i. We also show that {point, line}-domestic collineations are either point-domestic or line-domestic, and then we nail down the structure of the fixed elements of point-domestic collineations and of line-domestic collineations. We also show that {i, i + 1}-domestic collineations are either i-domestic or (i + 1)-domestic (under the assumption that i + 1 is not the type of the maximal subspaces if i is even). For polar spaces of rank 3, we obtain a full classification of all chamber-domestic collineations. All our results hold in the general case (finite or infinite) and generalize the full classification of all domestic collineations of polar spaces of rank 2 performed in Temmermans et al. (to appear in Ann Comb).

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Characterizations of Trialities of Type $$\mathrm {I}_{\mathsf {id}}$$ in Buildings of Type $$\mathsf {D}_4$$

Maldeghem

2014

Groups of Exceptional Type, Coxeter Groups and Related Geometries

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Domesticity in Generalized Quadrangles

Cited by 9 publications

References 11 publications

Symplectic polarities of buildings of type E 6

Symplectic polarities of buildings of type E 6

Collineations of polar spaces with restricted displacements

Characterizations of Trialities of Type $$\mathrm {I}_{\mathsf {id}}$$ in Buildings of Type $$\mathsf {D}_4$$

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