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Title:  Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic 
Author(s):  Wenzel, Christian 
Doctoral Committee Chair(s):  Haboush, William J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\Phi$, and a set of simple roots $\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties. In my thesis I generalize to subgroupschemes containing B. They are groupschemes, but not necessarily varieties; their algebras of functions might have nilpotent elements, i.e. they might not be reduced. In my thesis I show that in characteristic p $>$ 0, there are infinitely many whenever G $\not=$ 1, I exhibit their structure, and I classify them. I show that in characteristic p $>$ 3, the subgroupschemes containing B correspond to $\tilde\Delta$, the set of all maps from $\Delta$ to $\rm I\!N \cup \{\infty\},$ in such a way that it extends the classical classification of parabolic subgroups in terms of subsets of $\Delta$. To each $\varphi$ there is a parabolic P$\sb\varphi$ with $\rm P\sb\varphi = U\sb\varphi\cdot P\sb{I(\varphi)},$ I$(\varphi) = \{\alpha\in\Delta\mid\varphi(\alpha)=\infty\}$, $\rm P\sb{I(\varphi)} = (P\sb\varphi)\sb{red}$ = Spec(K (P$\sb\varphi$) /nilrad), U$\sb\varphi$ being a certain local unipotent subgroupscheme. In characteristic 2,3 the situation is more complicated. Furthermore I give a construction of G/P also for nonreduced P. I show that G/P is a rational projective variety, whenever char (K) $>$ 3. 
Issue Date:  1990 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/22442 
Rights Information:  Copyright 1990 Wenzel, Christian 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9114457 
OCLC Identifier:  (UMI)AAI9114457 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois