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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



$ {\mathrm A}_3$-proof of structure theorems for Chevalley groups of types $ {\mathrm E}_6$ and $ {\mathrm E}_7$ II. Main lemma

Author: N. A. Vavilov
Translated by: The author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 6.
Journal: St. Petersburg Math. J. 23 (2012), 921-942
MSC (2010): Primary 20G15, 20G35
Published electronically: September 17, 2012
MathSciNet review: 2962179
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Abstract: The author and Mikhail Gavrilovich proposed a geometric proof of the structure theorems for Chevalley groups of types $ \Phi = \mathrm {E}_6, \mathrm {E}_7$, based on the following fact. There are nontrivial root unipotents of type $ \mathrm {A}_2$ stabilizing columns of a root element. In the present paper it is shown that two adjacent columns of a root element can be stabilized simultaneously by a nontrivial root unipotent of type $ \mathrm {A}_3$. This makes it possible to prove structure theorems for Chevalley groups of types $ \mathrm {E}_6$ and $ \mathrm {E}_7$ and their forms, by using only the presence of split classical subgroups of very small ranks.

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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia

Keywords: Chevalley groups, elementary subgroups, normal subgroups, standard description, minimal modules, parabolic subgroups, decomposition of unipotents, root elements, orbits of highest weight vector, the Proof from the Book
Received by editor(s): May 25, 2010
Published electronically: September 17, 2012
Additional Notes: The present work was carried out in the framework of the RFBR projects 08-01-00756 “Decompositions of algebraic groups and their applications in representation theory and $K$-theory” and 10-01-90016 “The study of structure of forms of reductive groups and behavior of small unipotent elements in representations of algebraic groups”. Furthermore, at the final stage the author was supported by the RFBR projects 09-01-00762, 09-01-00784, 09-01-00878, 09-01-91333, 09-01-90304, and 10-01-92651.
Article copyright: © Copyright 2012 American Mathematical Society

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