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

${\mathrm A}_3$ -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
DOI: https://doi.org/10.1090/S1061-0022-2012-01223-9
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.

1. Eiichi Abe, Automorphisms of Chevalley groups over commutative rings, Algebra i Analiz 5 (1993), no. 2, 74–90 (Russian); English transl., St. Petersburg Math. J. 5 (1994), no. 2, 287–300. MR 1223171
2. Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
3. Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
4. N. A. Vavilov, How is one to view the signs of structure constants?, Algebra i Analiz 19 (2007), no. 4, 34–68 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519–543. MR 2381932, https://doi.org/10.1090/S1061-0022-08-01008-X
5. N. A. Vavilov, Numerology of quadratic equations, Algebra i Analiz 20 (2008), no. 5, 9–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 5, 687–707. MR 2492358, https://doi.org/10.1090/S1061-0022-09-01068-1
6. N. A. Vavilov and M. R. Gavrilovich, 𝐴₂-proof of structure theorems for Chevalley groups of types 𝐸₆ and 𝐸₇, Algebra i Analiz 16 (2004), no. 4, 54–87 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 4, 649–672. MR 2090851, https://doi.org/10.1090/S1061-0022-05-00871-X
7. N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, The structure of Chevalley groups: a proof from The Book, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 13, 36–76, 271–272 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 5, 626–645. MR 2253566, https://doi.org/10.1007/s10958-007-0003-y
8. N. A. Vavilov and V. G. Kazakevich, Yet another variation on the theme of decomposition of transvections, Vestnik St. Petersburg Univ. Math. 41 (2008), no. 4, 345–347. MR 2485399, https://doi.org/10.3103/S1063454108040092
9. N. A. Vavilov and V. G. Kazakevich, Decomposition of transvections for automorphisms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 365 (2009), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 18, 47–62, 262–263 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 161 (2009), no. 4, 483–491. MR 2749134, https://doi.org/10.1007/s10958-009-9578-9
10. N. A. Vavilov and V. G. Kazakevich, More variations on the decomposition of transvections, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 375 (2010), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 19, 32–47, 209–210 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 3, 322–330. MR 2749273, https://doi.org/10.1007/s10958-010-0137-1
11. N. A. Vavilov and A. Yu. Luzgarev, The normalizer of Chevalley groups of type 𝐸₆, Algebra i Analiz 19 (2007), no. 5, 37–64 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 5, 699–718. MR 2381940, https://doi.org/10.1090/S1061-0022-08-01016-9
12. -, The normalizer of Chevalley groups of type $\mathrm E_7$ (to appear). (Russian)
13. N. A. Vavilov and A. Yu. Luzgarev, A Chevalley group of type 𝐸₇ in the 56-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), no. Voprosy Teorii Predstvaleniĭ Algebr i Grupp. 20, 5–99, 288 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 180 (2012), no. 3, 197–251. MR 2784131, https://doi.org/10.1007/s10958-011-0641-y
14. -, $\mathrm A_2$ -proof of structure theorems for Chevalley groups of type $\mathrm E_8$ (to appear). (Russian)
15. N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, A Chevalley group of type 𝐸₆ in the 27-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 14, 5–68, 261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 145 (2007), no. 1, 4697–4736. MR 2354606, https://doi.org/10.1007/s10958-007-0304-1
16. N. A. Vavilov and S. I. Nikolenko, 𝐴₂-proof of structure theorems for a Chevalley group of type 𝐹₄, Algebra i Analiz 20 (2008), no. 4, 27–63 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 4, 527–551. MR 2473743, https://doi.org/10.1090/S1061-0022-09-01060-7
17. N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, Calculations in Chevalley groups over commutative rings, Dokl. Akad. Nauk SSSR 307 (1989), no. 4, 788–791 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 1, 145–147. MR 1020667
18. N. A. Vavilov and A. K. Stavrova, Basic reductions in the problem of the description of normal subgroups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 16, 30–52, 242–243 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 3, 2949–2960. MR 2742853, https://doi.org/10.1007/s10958-008-9019-1
19. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
20. A. Yu. Luzgarev, On overgroups of 𝐸(𝐸₆,𝑅) and 𝐸(𝐸₇,𝑅) in minimal representations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), no. Vopr. Teor. Predst. Algebr. i Grupp. 11, 216–243, 302 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 134 (2006), no. 6, 2558–2571. MR 2117858, https://doi.org/10.1007/s10958-006-0127-5
21. A. Yu. Luzgarëv, Description of the overgroups 𝐹₄ in 𝐸₆ over a commutative ring, Algebra i Analiz 20 (2008), no. 6, 148–185 (Russian); English transl., St. Petersburg Math. J. 20 (2009), no. 6, 955–981. MR 2530897, https://doi.org/10.1090/S1061-0022-09-01080-2
22. -, Overgroups of exceptional groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2008, pp. 1-106. (Russian)
23. I. M. Pevzner, The geometry of root elements in groups of type 𝐸₆, Algebra i Analiz 23 (2011), no. 3, 261–309 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 3, 603–635. MR 2896171, https://doi.org/10.1090/S1061-0022-2012-01210-0
24. I. M. Pevzner, The geometry of root elements in groups of type 𝐸₆, Algebra i Analiz 23 (2011), no. 3, 261–309 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 3, 603–635. MR 2896171, https://doi.org/10.1090/S1061-0022-2012-01210-0
25. T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713
26. Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
27. M. S. Tulenbaev, The Schur multiplier of the group of elementary matrices of finite order, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 86 (1979), 162–169, 191–192 (Russian). Algebraic numbers and finite groups. MR 535488
28. Eiichi Abe, Chevalley groups over local rings, Tôhoku Math. J. (2) 21 (1969), 474–494. MR 0258837, https://doi.org/10.2748/tmj/1178242958
29. Eiichi Abe, Chevalley groups over commutative rings, Radical theory (Sendai, 1988) Uchida Rokakuho, Tokyo, 1989, pp. 1–23. MR 999577
30. Eiichi Abe, Normal subgroups of Chevalley groups over commutative rings, Algebraic 𝐾-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 1–17. MR 991973, https://doi.org/10.1090/conm/083/991973
31. Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185–198. MR 0439947, https://doi.org/10.2748/tmj/1178240833
32. R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 0318337
33. Roger W. Carter, Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR 0407163
34. W. van der Kallen, Another presentation for Steinberg groups, Nederl. Akad. Wetensch. Proc. Ser. A 80=Indag. Math. 39 (1977), no. 4, 304–312. MR 0463263
35. Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 0240214
36. Victor Petrov and Anastasia Stavrova, The Tits indices over semilocal rings, Transform. Groups 16 (2011), no. 1, 193–217. MR 2785501, https://doi.org/10.1007/s00031-010-9112-7
37. Eugene Plotkin, On the stability of the 𝐾₁-functor for Chevalley groups of type 𝐸₇, J. Algebra 210 (1998), no. 1, 67–85. MR 1656415, https://doi.org/10.1006/jabr.1998.7535
38. Eugene Plotkin, Andrei Semenov, and Nikolai Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61–95. MR 1492062, https://doi.org/10.1142/S0218196798000053
39. Michael R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965–1004. MR 0322073, https://doi.org/10.2307/2373742
40. Michael R. Stein, Stability theorems for 𝐾₁, 𝐾₂ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77–108. MR 528869
41. Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, 𝐾-Theory 19 (2000), no. 2, 109–153. MR 1740757, https://doi.org/10.1023/A:1007853629389
42. Nikolai A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 219–335. MR 1150262
43. Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201–250. MR 1809357
44. N. A. Vavilov, Do it yourself structure constants for Lie algebras of types 𝐸_{𝑙}, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), no. Vopr. Teor. Predst. Algebr. i Grupp. 8, 60–104, 281 (English, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1513–1548. MR 1875718, https://doi.org/10.1023/B:JOTH.0000017882.04464.97
45. Nikolai Vavilov, An 𝐴₃-proof of structure theorems for Chevalley groups of types 𝐸₆ and 𝐸₇, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283–1298. MR 2355697, https://doi.org/10.1142/S0218196707003998
46. N. Vavilov, A. Luzgarev, and A. Stepanov, Calculations in exceptional groups over rings, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 373 (2009), no. Teoriya Predstavleniĭ, Dinamicheskie Sistemy, Kombinatornye Metody. XVII, 48–72, 346 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 168 (2010), no. 3, 334–348. MR 2749253, https://doi.org/10.1007/s10958-010-9984-z
47. Nikolai Vavilov and Eugene Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), no. 1, 73–113. MR 1409655, https://doi.org/10.1007/BF00047884
48. N. Vavilov, A closer look at weight diagrams of types $({\mathrm E}_6,\varpi _1)$ and $({\mathrm E}_7,\varpi _7)$ (2012) (to appear).
49. N. A. Vavilov and A. V. Shchegolev, Overgroups of subsystem subgroups in exceptional groups: levels (2012) (to appear).
50. N. A. Vavilov and Z. Zhang, Subnormal subgroups of Chevalley groups. I. Cases ${\mathrm E}_6$ and ${\mathrm E}_7$ (2012) (to appear).
51. M. Wendt, On homotopy invariance for homology of rank two groups (2012) (to appear).