A characterization of Lie incidence geometries of exceptional type E - PDF document

A characterization of Lie incidence geometries of exceptional type E 7 , 4 and E 8 , 4 Silvia Onofrei Mathematics Department, Kansas State University 137 Cardwell Hall, Manhattan KS 66506 email: onofrei@math.ksu.edu 1

2 Let D be one of the following Dynkin diagrams: ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1 2 3 5 6 7 1 2 3 5 6 7 8 ◦ ◦ 4 4 Label the node 4 by P , points, and the node 3 by L , lines. We get in this way the Lie incidence geometries (Cooperstein, [8]) of exceptional type E 7 , 4 and E 8 , 4 . The object of this paper is the following: The Main Theorem. Let ∆ be a parapolar space with thick lines which is locally A n, 3 , n = 6 , 7 . Then ∆ is the homo- morphic image of a building of type E 7 or E 8 . We start with a point-line geometry, ∆ = ( P , L ). Our approach is based on the local assumption that the residue of a point in ∆ is a ge- ometry of Grassmann type A n, 3 , n = 6 , 7 and on the global assumption that ∆ is a parapolar space. Cooperstein theory provides us with two classes of maximal singular subspaces, which will be denoted ¯ A and ¯ B and with a class of symplecta S , nondegenerate polar spaces of type D 4 . First, we prove that there is a class of 2-convex subspaces, D , each one isomorphic to D 5 , 5 ( K ), for some division ring K , and such that each symplecton S ∈ S is contained in a unique member D ( S ) ∈ D . ◦ ◦ ◦ ◦ ◦ ◦ ◦ � � � � � ¯ ¯ ¯ ¯ B L A B L A D ◦ ◦ P P Now, let Γ be a locally D -truncated geometry over K = ( P , L , ¯ A , ¯ B , D ). A residually connected sheaf over all nonempty flags of Γ is constructed. Then ∆ is the homomorphic image of a building geometry belonging to the diagram D and whose truncation to K is isomorphic to Γ.

3 1. Preliminaries and definitions A geometry Γ is residually connected if and only if: (i) every flag of corank one lies in a maximal flag; (ii) any flag of corank at least two has a connected residue. Let f : ¯ Γ → Γ be an epimorphism of geometries over I . Then f is said to be a full epimorphism if whenever x ∼ y in Γ, there exists some y in ¯ x ∈ f − 1 ( x ) and some ¯ y ∈ f − 1 ( y ) with ¯ ¯ x ∼ ¯ Γ. ( G ) A point-line geometry Γ = ( P , L ) is a Gamma space if for x ∈ P and L ∈ L with | x ⊥ ∩ L | > 1 then x ∈ L ⊥ . A parapolar space is a point-line geometry Γ = ( P , L ) which is defined by the following axioms: 1. Γ is a connected gamma space. 2. For any L ∈ L , L ⊥ is not a clique in the point-collinearity graph. 3. For any pair x, y ∈ P , d ( x, y ) = 2, x ⊥ ∩ y ⊥ is empty, a point or a nondegenerate polar space of rank at least 2. If | x ⊥ ∩ y ⊥ | = 1 then ( x, y ) is called a special pair , if | x ⊥ ∩ y ⊥ | > 1 then ( x, y ) is a polar pair . If no special pairs exist, Γ is called a strong parapolar space . It was proved by Cooperstein [9] that whenever ( x, y ) is a polar pair, i.e. x ⊥ ∩ y ⊥ is a nondegenerate polar space of rank k , the convex clo- sure of x and y , denoted by < x, y > , is a non-degenerate polar space of rank k + 1, which is called a symplecton . We write S = < x, y > . The family of all symplecta in Γ will be denoted by S . The parapolar spaces are partial linear spaces whose singular subspaces are all projec- tive. Every singular subspace of rank at most 2 lies in a symplecton. Given S ∈ S , x ∈ P \ S then x ⊥ ∩ S is a singular, possibly emtpy, subspace of S .

4 2. Some properties of A n, 3 Let V be a n + 1-dimensional vector space over some division ring K . Define a point-line geometry Γ = ( P , L ) as follows: the points of Γ are the 3-dimensional subspaces of V , the lines are the (2 , 4)-flags of V . Then Γ is a Grassmann space of type A n, 3 . In the sequel we restrict ourselves to the case n = 6 , 7. There are two classes of maximal singular subspaces: (1) The class A , which are PG (3)’s; (2) The class B , which are PG ( n − 2)’s. Let M = A ∪ B be the collection of all maximal singular subspaces of Γ. If A ∈ A , B ∈ B then A ∩ B is empty or a projective line. If A, A ′ ∈ A ( B, B ′ ∈ B ) then A ∩ A ′ ( B ∩ B ′ ) is empty or a point. The Grassmann spaces are strong parapolar spaces. Their symplecta are polar spaces of rank 3 and type D 3 , they are (1 , 5)-flags of V . The point-collinearity graph has diameter 3. (1) If x ∈ P , S ∈ S , x �∈ S then x ⊥ ∩ S is empty, a point or a plane. (2) If S 1 , S 2 ∈ S then S 1 ∩ S 2 can be empty, a point, a line or a common maximal singular subspace of the two symplecta. (3) If S ∈ S and M ∈ M then S ∩ M is empty, a point or a plane. In [13] Shult gave a characterization in terms of points and maximal singular subspaces, of certain parapolar spaces, including some Grass- manianns. We will use his results here, in order to formulate the fol- lowing property of the Grassmann spaces A 6 , 3 and A 7 , 3 : ( D ) For any maximal singular subspace M ∈ M and point p ∈ P \ M , x ⊥ ∩ M is empty or a line.

5 3. The class D of subspaces In what follows Γ = ( P , L ) will be a parapolar space which is locally A n, 3 ( K ) for n = 6 , 7 and some division ring K (i.e. the residue of every point x ∈ P is a geometry of type A n, 3 , n = 6 , 7). (L.1) There are two classes of maximal singular subspaces ¯ A (which are PG (4)’s) and ¯ B (which are PG ( n − 1)’s, n = 6 , 7). Two maximal singular subspaces from the same class meet at the empty set or a line. Two maximal singular subspaces which belong to different classes meet at the empty set, a point or a plane. (L.2) Let ¯ A i ∈ ¯ A , ¯ B i ∈ B , i = 1 , 2, S ∈ S and denote by A i = ¯ A i ∩ S , B i = ¯ B i ∩ S . Then A i ∩ B i can be a point or a plane and A 1 ∩ A 2 (or B 1 ∩ B 2 ) can be empty, a line or they are equal. In [13] Shult gave a characterization in terms of points and maximal singular subspaces, of certain parapolar spaces, including some Grass- manianns. We will use his results here, in order to formulate: M then x ⊥ ∩ ¯ (L.3) Given ¯ M ∈ ¯ M and x ∈ P \ ¯ M is empty, a point or a plane. (L.4) Given S ∈ S , x ∈ P \ S then x ⊥ ∩ S is empty, a point, a line or a maximal singular subspace of S . Let S ∈ S . The two classes of maximal singular subspaces are: M 4 ( S ) = { A | A = ¯ A ∩ S for ¯ A ∈ ¯ A} M n − 1 ( S ) = { B | B = ¯ B ∩ S for ¯ B ∈ ¯ B} , n = 6 , 7 Then, we define: N 4 ( S ) = { x ∈ P | x ⊥ ∩ S ∈ M 4 ( S ) } D ( S ) := S ∪ N 4 ( S )

6 Theorem. Let Γ = ( P , L ) be a parapolar space which is locally A n, 3 , n = 6 , 7 . There is a collection D of geodesically closed subspaces, each one isomorphic to D 5 , 5 ( K ) , for some division ring K such that if S ∈ S then there is a unique member D ( S ) ∈ D containing S . In what follows, we will use the following notations: for S ∈ S and a point p ∈ N 4 ( S ) we will denote by A p = p ⊥ ∩ S ∈ M 4 ( S ) and by A p = < p, A p > ∈ ¯ ¯ A . Proposition 1. Let S ∈ S , then D ( S ) is a subspace of Γ . Proof. Let x, y ∈ D ( S ) be two collinear points. If x, y ∈ S we are done since S is a geodesically closed subspace of Γ. If x ∈ N 4 ( S ) and y ∈ S then the line L 1 = xy ⊂ ¯ A x ⊂ D ( S ). Thus we may assume that the line L 1 = xy is such that L 1 ∩ S = ∅ . A x , A y are maximal singular subspaces of S belonging to the same class. Thus there are three possi- ble relations among them: A x = A y , A x ∩ A y = L a line or A x ∩ A y = ∅ . a) Assume first that A x = A y . Let z ∈ L 1 \ { x, y } . Since A x = A y ⊂ A x ∩ ¯ ¯ A y , by (L.2) it follows that ¯ A x = ¯ A y . This means that L 1 ⊂ ¯ A x and thus z ∈ D ( S ). b) Assume now that A x ∩ A y = L ∈ L ( S ) and let z ∈ L 1 \ { x, y } . As L ⊂ z ⊥ ∩ S we can have z ⊥ ∩ S is a line or a Then L ⊂ z ⊥ . maximal singular subspace of S . Let w ∈ A y \ z ⊥ . Let π = w ⊥ ∩ A x . Now B = < w, π > ∈ M n − 1 ( S ). This is true since B ∩ A x = π and B ∩ A y = < w, L > , both planes in S . Therefore, B and A x , A y belong to different classes of maximal singular subspaces of S . Let R = < w, x > . Note that y ∈ R thus z ∈ R . Also B is a maximal singular subspace of R . In R , z ⊥ ∩ B = π z (a plane) and therefore, by ( L. 4), z ⊥ ∩ S = A z a maximal singular subspace of S . Thus, we have proved that if z ⊥ ∩ S ⊃ L then z ⊥ ∩ S must be a maximal sin- gular subspace of S . It remains to show that A z ∈ M 4 ( S ). Claim :