Extensions of isomorphisms for a ‰ ne Grassmannians over F 2

In Blok [1] a‰nely rigid classes of geometries were studied. These are classes B of geometries with the following property: Given any two geometries G1;G2 A B with subspaces S1 and S2 respectively, then any isomorphism G1 S1 ! G2 S2 uniquely extends to an isomorphism G1 ! G2. Suppose G belongs to an a‰nely rigid class. Then for any subspace S we have AutðG SÞcAutðGÞ. Suppose that, in addition, G is embedded into the projective space PðVÞ for some vector space V . Then one may think of V as a ‘‘natural’’ embedding if every automorphism of G is induced by some (semi-) linear automorphism of V . This is for instance true of the projective geometry G 1⁄4 PðVÞ itself by the fundamental theorem of projective geometry. Clearly since G belongs to an a‰nely rigid class and has a natural embedding into PðVÞ, also the embedding G S into PðVÞ is natural. In Blok [1] the notion of a layer-extendable class was introduced and it was shown that layer-extendable classes are a‰nely 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 a‰ne projective geometries (1-Grassmannians, if you will) are simply complete graphs, clearly they are not a‰nely rigid at all. In the present note we show that also the class of 2-Grassmannians over F2 fails to form an a‰nely rigid class, although in a less dramatic way, whereas the class of k-Grassmannians of projective spaces of dimension n over F2 where 3c kc n 2 is in fact a‰nely rigid.