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Lie Incidence Systems from Projective Varieties

Authors: Arjeh M. Cohen and Bruce N. Cooperstein
Journal: Proc. Amer. Math. Soc. 126 (1998), 2095-2102
MSC (1991): Primary 51B25; Secondary 14L17, 14M15
MathSciNet review: 1443819
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Abstract: The homogeneous space $G/P_{\lambda }$, where $G$ is a simple algebraic group and $P_{\lambda }$ a parabolic subgroup corresponding to a fundamental weight $\lambda $ (with respect to a fixed Borel subgroup $B$ of $G$ in $P_{\lambda }$), is known in at least two settings. On the one hand, it is a projective variety, embedded in the projective space corresponding to the representation with highest weight $\lambda $. On the other hand, in synthetic geometry, $G/P_{\lambda }$ is furnished with certain subsets, called lines, of the form $gB\langle r\rangle P_{\lambda }/P_{\lambda }$ where $r$ is a preimage in $G$ of the fundamental reflection corresponding to $\lambda $ and $g\in G$. The result is called the Lie incidence structure on $G/P_{\lambda }$. The lines are projective lines in the projective embedding. In this paper we investigate to what extent the projective variety data determines the Lie incidence structure.

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Additional Information

Arjeh M. Cohen
Affiliation: Fac. Wisk. en Inf., TUE Postbus 513, 5600 MB Eindhoven, The Netherlands

Bruce N. Cooperstein
Affiliation: Fac. Wisk. en Inf., TUE Postbus 513, 5600 MB Eindhoven, The Netherlands; Department of Mathematics, University of California, Santa Cruz, California 95064

Keywords: Groups of Lie type, Lie incidence systems, geometry, quadrics
Received by editor(s): July 6, 1996
Received by editor(s) in revised form: December 18, 1996
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society