𝐴₃-proof of structure theorems for Chevalley groups of types 𝐸₆ and 𝐸₇ II. Main lemma

Vavilov, N.

doi:10.1090/S1061-0022-2012-01223-9

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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
Posted: September 17, 2012
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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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