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 Algebra i Analiz, 2008, Volume 20, Issue 4, Pages 27–63 (Mi aa521)

Research Papers

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

N. A. Vavilova, S. I. Nikolenkob

a St. Petersburg State University, Department of Mathematics and Mechanics
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

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 reduction in 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 required the 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 embedding of $\mathrm{A}_2$. Unlike all previous proofs, neither results pertaining to the field case, nor explicit knowledge of structure constants and defining equations are 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.

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

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English version:
St. Petersburg Mathematical Journal, 2009, 20:4, 527–551

Bibliographic databases:

MSC: 20G15, 20G35

Citation: N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, Algebra i Analiz, 20:4 (2008), 27–63; St. Petersburg Math. J., 20:4 (2009), 527–551

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation”, J. Math. Sci. (N. Y.), 145:1 (2007), 4697–4736
2. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, J. Math. Sci. (N. Y.), 140:5 (2007), 626–645
3. N. A. Vavilov, A. K. Stavrova, “Basic reductions for the description of normal subgroups”, J. Math. Sci. (N. Y.), 151:3 (2008), 2949–2960
4. N. A. Vavilov, “Can one see the signs of structure constants?”, St. Petersburg Math. J., 19:4 (2008), 519–543
5. N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718
6. N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948
7. Vavilov N., “An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$”, Internat. J. Algebra Comput., 17:5–6 (2007), 1283–1298
8. N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707
9. A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981
10. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95
11. A. S. Ananievskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$”, J. Math. Sci. (N. Y.), 161:4 (2009), 461–473
12. Bak A., Hazrat R., Vavilov N., “Localization-completion strikes again: relative $K_1$ is nilpotent by abelian”, J. Pure Appl. Algebra, 213:6 (2009), 1075–1085
13. J. Math. Sci. (N. Y.), 168:3 (2010), 334–348
14. Hazrat R., Vavilov N., “Bak's work on the $K$-theory of rings”, J. K-Theory, 4:1 (2009), 1–65
15. N. A. Vavilov, V. G. Kazakevich, “More variations on decomposition of transvections”, J. Math. Sci. (N. Y.), 171:3 (2010), 322–330
16. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27
17. N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251
18. I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635
19. N. A. Vavilov, “An $\mathrm A_3$-proof of the structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$. II. The main lemma”, St. Petersburg Math. J., 23:6 (2012), 921–942
20. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550
21. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195
22. Hazrat R. Vavilov N. Zhang Z., “Relative Commutator Calculus in Chevalley Groups”, J. Algebra, 385 (2013), 262–293
23. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369
24. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
25. Preusser R., “Sandwich Classification For O2N+1(R) and U2N+1(R, Delta) Revisited”, J. Group Theory, 21:4 (2018), 539–571
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