Extensions of isomorphisms for affine Grassmannians over

Abstract

Abstract In Blok [R. J. Blok, Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces. Adv. Geom. 5 (2005), 509–532.] affinely rigid classes of geometries were studied. These are classes B of geometries with the following property: Given any two geometries Γ1, Γ2 ∈ B with subspaces 𝒮1 and 𝒮2 respectively, then any isomorphism Γ1 − 𝒮1 → Γ2 − 𝒮2 uniquely extends to an isomorphism Γ1 → Γ2. Suppose Γ belongs to an affinely rigid class. Then for any subspace 𝒮 we have Aut(Γ − 𝒮) ≤ Aut(Γ). Suppose that, in addition, Γ is embedded into the projective space ℙ(V) for some vector space V. Then one may think of V as a “natural” embedding if every automorphism of Γ is induced by some (semi-) linear automorphism of V. This is for instance true of the projective geometry Γ = ℙ(V) itself by the fundamental theorem of projective geometry. Clearly since Γ belongs to an affinely rigid class and has a natural embedding into ℙ(V), also the embedding Γ − 𝒮 into ℙ(V) is natural. In Blok [R. J. Blok, Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces. Adv. Geom. 5 (2005), 509–532.] the notion of a layer-extendable class was introduced and it was shown that layer-extendable classes are affinely rigid. As an application, it was shown that the union of most projective geometries, (dual) polar spaces, and strong parapolar spaces forms an affinely rigid class. However, the geometries motivating that study, the Grassmannians defined over 𝔽2, were not included in this class because they do not form a layer-extendable class. Since affine projective geometries (1-Grassmannians, if you will) are simply complete graphs, clearly they are not affinely rigid at all. In the present note we show that also the class of 2-Grassmannians over 𝔽2 fails to form an affinely rigid class, although in a less dramatic way, whereas the class of k-Grassmannians of projective spaces of dimension n over 𝔽2 where 3 ≤ k ≤ n − 2 is in fact affinely rigid.