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Adjacency preserving mappings
of invariant subspaces of a null system

Author: Wen-ling Huang
Journal: Proc. Amer. Math. Soc. 128 (2000), 2451-2455
MSC (1991): Primary 51A50; Secondary 51B25
Published electronically: November 29, 1999
MathSciNet review: 1690993
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Abstract: In the space $I_r$ of invariant $r$-dimensional subspaces of a null system in $(2r+1)$-dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections $\varphi:I_r\to I_r$, for which both $\varphi$ and $\varphi^{-1}$ preserve adjacency. In the present paper we show that the two conditions $\varphi:I_r\to I_r$ is a surjection and $\varphi$ preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.

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  • 1. R. Baer. Linear Algebra and Projective Geometry. Academic Press, New York, San Francisco, London, 1952. MR 14:675j
  • 2. Walter Benz, Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin-New York, 1973 (German). Geometrien von Möbius, Laguerre-Lie, Minkowski in einheitlicher und grundlagengeometrischer Behandlung; Die Grundlehren der mathematischen Wissenschaften, Band 197. MR 0353137
  • 3. Walter Benz, Geometrische Transformationen, Bibliographisches Institut, Mannheim, 1992 (German). Unter besonderer Berücksichtigung der Lorentztransformationen. [With special reference to the Lorentz transformations]. MR 1183223
  • 4. W.-L. Chow. On the geometry of algebraic homogeneous spaces. Ann. Math., 50(1):32-67, 1949. MR 10:396d
  • 5. L.K. Hua. Geometries of matrices III. Fundamental theorems in the geometries of symmetric matrices. Trans. Amer. Math. Soc., 61:229-255, 1947. MR 9:171e
  • 6. L.K. Hua. Geometries of symmetric matrices over any field with characteristic other than two. Ann. Math., 50:8-31, 1949. MR 10:424h
  • 7. W.-l. Huang. Adjacency preserving mappings of Grassmann spaces. Abh. Math. Sem. Univ. Hamburg, 68:65-77, 1998. CMP 99:05
  • 8. June A. Lester, Distance preserving transformations, Handbook of incidence geometry, North-Holland, Amsterdam, 1995, pp. 921–944. MR 1360731,
  • 9. Zhe-xian Wan, Geometry of matrices, Progress in algebraic combinatorics (Fukuoka, 1993) Adv. Stud. Pure Math., vol. 24, Math. Soc. Japan, Tokyo, 1996, pp. 443–453. MR 1414475,
  • 10. Zhe-Xian Wan, Geometry of matrices, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. In memory of Professor L. K. Hua (1910–1985). MR 1409610

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Additional Information

Wen-ling Huang
Affiliation: Mathematisches Seminar, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Keywords: Null system, adjacency preserving mappings, symmetric matrices, Lie transformations
Received by editor(s): September 25, 1998
Published electronically: November 29, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society