Base subsets of symplectic Grassmannians
Mark Pankov
University of Warmia and Mazury Department of Mathematics and Information Technology Żolnierska 14A 10-561 Olsztyn Poland
DOI: 10.1007/s10801-006-0051-2
Abstract
Let V and V$^{\prime}$ be 2 n-dimensional vector spaces over fields F and F$^{\prime}$. Let also Ω : V\times V\rightarrow F and Ω $^{\prime}$: V$^{\prime}$\times V$^{\prime}$\rightarrow F$^{\prime}$ be non-degenerate symplectic forms. Denote by Π and Π $^{\prime}$ the associated (2 n - 1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π $^{\prime}$ will be denoted by G k {\mathcal G}_{k} and G \textcent k {\mathcal G}'_{k}, respectively. Apartments of the associated buildings intersect G k {\mathcal G}_{k} and G \textcent k {\mathcal G}'_{k} by so-called base subsets. We show that every mapping of G k {\mathcal G}_{k} to G \textcent k {\mathcal G}'_{k} sending base subsets to base subsets is induced by a symplectic embedding of Π to Π $^{\prime}$.
Pages: 143–159
Keywords: keywords Tits building; symplectic grassmannians; base subsets
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References
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2. W.L. Chow, “On the geometry of algebraic homogeneous spaces,” Ann. Math. 50 (1949), 32-67.
3. C.A. Faure and A. Fr\ddot olicher, “Morphisms of projective geometries and semilinear maps,” Geom. Dedicata 53 (1994), 237-262.
4. C.A. Faure, “An elementary proof of the fundamental theorem of projective geometry,” Geom. Dedicata 90 (2002) 145-151.
5. H. Havlicek, “A generalization of Brauner's theorem on linear mappings,” Mitt. Math. Sem. Univ. Giessen 215 (1994), 27-41.
6. W.-l. Huang and A. Kreuzer, “Basis preserving maps of linear spaces,” Arch. Math. (Basel) 64(6) (1995), 530-533.
7. A. Kreuzer, “On the definition of isomorphisms of linear spaces,” Geom. Dedicata 61 (1996), 279-283.
8. M. Pankov, “Transformations of Grassmannians and automorphisms of classical groups,” J. Geom. 75 (2002), 132-150.
9. M. Pankov, “A characterization of geometrical mappings of Grassmann spaces,” Results Math. 45 (2004), 319-327.
10. M. Pankov, “Mappings of the sets of invariant subspaces of null systems,” Beitr\ddot age zur Algebra Geom. 45 (2004), 389-399.
11. M. Pankov, K. Pra\?zmovski, and M. \?Zynel, “Geometry of polar Grassmann spaces,” Demonstratio Math. 39 (2006), 625-637.
12. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer, Berlin, 1974.
13. H. Van Maldeghem, Private communication.