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$ \mathrm A_2$-proof of structure theorems for Chevalley groups of type $ \mathrm F_4$


Authors: N. A. Vavilov and S. I. Nikolenko
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 527-551
MSC (2000): Primary 20G15, 20G35
DOI: https://doi.org/10.1090/S1061-0022-09-01060-7
Published electronically: June 1, 2009
MathSciNet review: 2473743
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Abstract: A new geometric proof is given for the standard description of subgroups in the Chevalley group $ G=G(\mathrm{F}_4,R)$ of type $ \mathrm{F}_4$ over a commutative ring $ R$ that are normalized by the elementary subgroup $ E(\mathrm{F}_4,R)$. There are two major approaches to the proof of such results. Localization proofs (Quillen, Suslin, Bak) are based on a reduction in the dimension. The first proofs of this type for exceptional groups were given by Abe, Suzuki, Taddei and Vaserstein, but they invoked the Chevalley simplicity theorem and reduction modulo the radical. At about the same time, the first author, Stepanov, and Plotkin developed a geometric approach, decomposition of unipotents, based on reduction in the rank of the group. This approach combines the methods introduced in the theory of classical groups by Wilson, Golubchik, and Suslin with ideas of Matsumoto and Stein coming from representation theory and $ K$-theory. For classical groups in vector representations, the resulting proofs are quite straightforward, but their generalizations to exceptional groups require an explicit knowledge of the signs of action constants, and of equations satisfied by the orbit of the highest weight vector. They depend on the presence of high rank subgroups of types $ \mathrm{A}_l$ or $ \mathrm{D}_l$, such as $ \mathrm{A}_5\le\mathrm{E}_6$ and $ \mathrm{A}_7\le\mathrm{E}_7$. The first author and Gavrilovich introduced a new twist to the method of decomposition of unipotents, which made it possible to give an entirely elementary geometric proof (the proof from the Book) for Chevalley groups of types $ \Phi=\mathrm{E}_6,\mathrm{E}_7$. This new proof, like the proofs for classical cases, relies upon the embedding of $ \mathrm{A}_2$. Unlike all previous proofs, neither results pertaining to the field case nor an explicit knowledge of structure constants and defining equations is ever used. In the present paper we show that, with some additional effort, we can make this proof work also for the case of $ \Phi=\mathrm{F}_4$. Moreover, we establish some new facts about Chevalley groups of type $ \mathrm{F}_4$ and their 27-dimensional representation.


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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 20, Petrodvorets, 198504 St. Petersburg, Russia
Email: nikolai-vavilov@yandex.ru

S. I. Nikolenko
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 20, Petrodvorets, 198504 St. Petersburg, Russia

DOI: https://doi.org/10.1090/S1061-0022-09-01060-7
Keywords: Chevalley group, elementary subgroup, normal subgroups, standard description, minimal module, parabolic subgroups, decomposition of unipotents, root element, orbit of the highest weight vector, the proof from the Book
Received by editor(s): October 25, 2007
Published electronically: June 1, 2009
Additional Notes: Supported by the RFBR grants 03–01–00349 (POMI RAN) and INTAS 03-51-3251. Part of the work was carried out during the authors’ stay at the University of Bielefeld with the support of SFB-343 and INTAS 00–566
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