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Rieuwert J. Blok

Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces

Let B be a class of point-line geometries. Given ?_i ? B with subspace _i for i = 1, 2, does any isomorphism ?₁ ? ₁ ? ?₂ ? ₂ extend to a unique isomorphism ?₁ ? ?₂? It is known to be true if B is the class of almost all projective spaces or the class of almost all nondegenerate polar spaces. We show that this is true for the class of almost all strong parapolar spaces, including dual polar spaces.

A special case occurs when ?₁ = ?₂ = ? has an embedding into a projective space ?(V ) that is natural in the sense that Aut(? ) ? P?L(V ). Then the question becomes whether ?(V ) is also the natural embedding for ? ? . Our result shows that in most cases the stabilizer Stab_Aut(?)(? ? ) is faithful on ? ?  and equals Aut(? ? ) and so the answer is affirmative. We know that there exist some interesting exceptions. These will be covered in a subsequent paper.

Advances in Geometry, Walter de Gruyter

Print ISSN: 1615-715X
Volume: 5, 10/2005
Seiten: 509 - 532

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