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The theory of buildings, created by J. Tits three decads ago, has ooered a uniied geometric treatment of nite simple groups of Lie type, both of classical and of exceptional type. (See Tits 19] and 20] for an exposition of that theory; also Ronan 15] and Brown 1].) Diagram geometry (see 13] for an exposition) is a generalization of the theory of buildings.(More)
Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆ \ H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar(More)
In this paper we consider partial linear spaces containing a set of subspaces isomorphic to affine planes, such that the lines and these afline planes on a fixed point form a non-degenerate polar spaces of rank at least 2. We obtain a complete classification, provided that the rank is at least 3. The study of geometries on the absolute points of polarities(More)
Let 2 be a finite thick dual polar space of rank 3. We say that a hyperplane H of 2 is locally singular (respectively, quadrangular or ovoidal) if H & Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of 2. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally(More)
Let be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space , i.e., for every point x of , e maps the set of points of at non-maximal distance from x into a hyperplane e∗(x) of . Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full(More)
Cooperstein [6], [7] proved that every finite symplectic dual polar space DW (2n− 1, q), q = 2, can be generated by 2n n − 2n n−2 points and that every finite Hermitian dual polar space DH(2n − 1, q 2), q = 2, can be generated by 2n n points. In the present paper, we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces(More)