Skip to Main Content

St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

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

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

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

by N. A. Vavilov
Translated by: The author
St. Petersburg Math. J. 23 (2012), 921-942
DOI: https://doi.org/10.1090/S1061-0022-2012-01223-9
Published electronically: September 17, 2012
PDF | Request permission

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.
References
  • 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
  • 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
  • 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, DOI 10.1007/978-3-540-89394-3
  • 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, DOI 10.1090/S1061-0022-08-01008-X
  • 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, DOI 10.1090/S1061-0022-09-01068-1
  • N. A. Vavilov and M. R. Gavrilovich, $A_2$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, 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, DOI 10.1090/S1061-0022-05-00871-X
  • 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, DOI 10.1007/s10958-007-0003-y
  • 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, DOI 10.3103/S1063454108040092
  • 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, DOI 10.1007/s10958-009-9578-9
  • 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, DOI 10.1007/s10958-010-0137-1
  • N. A. Vavilov and A. Yu. Luzgarev, The normalizer of Chevalley groups of type $E_6$, 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, DOI 10.1090/S1061-0022-08-01016-9
  • —, The normalizer of Chevalley groups of type $\mathrm E_7$ (to appear). (Russian)
  • N. A. Vavilov and A. Yu. Luzgarev, A Chevalley group of type $E_7$ 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, DOI 10.1007/s10958-011-0641-y
  • —, $\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm E_8$ (to appear). (Russian)
  • N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, A Chevalley group of type $E_6$ 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, DOI 10.1007/s10958-007-0304-1
  • N. A. Vavilov and S. I. Nikolenko, $A_2$-proof of structure theorems for a Chevalley group of type $F_4$, 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, DOI 10.1090/S1061-0022-09-01060-7
  • 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
  • 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, DOI 10.1007/s10958-008-9019-1
  • E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
  • A. Yu. Luzgarev, On overgroups of $E(E_6,R)$ and $E(E_7,R)$ 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, DOI 10.1007/s10958-006-0127-5
  • A. Yu. Luzgarëv, Description of the overgroups $F_4$ in $E_6$ 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, DOI 10.1090/S1061-0022-09-01080-2
  • —, Overgroups of exceptional groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2008, pp. 1–106. (Russian)
  • I. M. Pevzner, The geometry of root elements in groups of type $\rm E_6$, 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, DOI 10.1090/S1061-0022-2012-01210-0
  • I. M. Pevzner, The geometry of root elements in groups of type $\rm E_6$, 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, DOI 10.1090/S1061-0022-2012-01210-0
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
  • Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
  • 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
  • Eiichi Abe, Chevalley groups over local rings, Tohoku Math. J. (2) 21 (1969), 474–494. MR 258837, DOI 10.2748/tmj/1178242958
  • Eiichi Abe, Chevalley groups over commutative rings, Radical theory (Sendai, 1988) Uchida Rokakuho, Tokyo, 1989, pp. 1–23. MR 999577
  • 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, DOI 10.1090/conm/083/991973
  • Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. (2) 28 (1976), no. 2, 185–198. MR 439947, DOI 10.2748/tmj/1178240833
  • R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
  • Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
  • 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
  • Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
  • Victor Petrov and Anastasia Stavrova, The Tits indices over semilocal rings, Transform. Groups 16 (2011), no. 1, 193–217. MR 2785501, DOI 10.1007/s00031-010-9112-7
  • Eugene Plotkin, On the stability of the $K_1$-functor for Chevalley groups of type $E_7$, J. Algebra 210 (1998), no. 1, 67–85. MR 1656415, DOI 10.1006/jabr.1998.7535
  • Eugene Plotkin, Andrei Semenov, and Nikolai Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61–95. MR 1492062, DOI 10.1142/S0218196798000053
  • Michael R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965–1004. MR 322073, DOI 10.2307/2373742
  • Michael R. Stein, Stability theorems for $K_{1}$, $K_{2}$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77–108. MR 528869, DOI 10.4099/math1924.4.77
  • Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, $K$-Theory 19 (2000), no. 2, 109–153. MR 1740757, DOI 10.1023/A:1007853629389
  • 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
  • Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201–250. MR 1809357
  • 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, DOI 10.1023/B:JOTH.0000017882.04464.97
  • Nikolai Vavilov, An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283–1298. MR 2355697, DOI 10.1142/S0218196707003998
  • 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, DOI 10.1007/s10958-010-9984-z
  • Nikolai Vavilov and Eugene Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), no. 1, 73–113. MR 1409655, DOI 10.1007/BF00047884
  • N. Vavilov, A closer look at weight diagrams of types $({\mathrm E}_6,\varpi _1)$ and $({\mathrm E}_7,\varpi _7)$ (2012) (to appear).
  • N. A. Vavilov and A. V. Shchegolev, Overgroups of subsystem subgroups in exceptional groups: levels (2012) (to appear).
  • N. A. Vavilov and Z. Zhang, Subnormal subgroups of Chevalley groups. I. Cases ${\mathrm E}_6$ and ${\mathrm E}_7$ (2012) (to appear).
  • M. Wendt, On homotopy invariance for homology of rank two groups (2012) (to appear).
Similar Articles
  • Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 20G15, 20G35
  • Retrieve articles in all journals with MSC (2010): 20G15, 20G35
Bibliographic Information
  • N. A. Vavilov
  • Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
  • Email: nikolai-vavilov@yandex.ru
  • 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.
  • © Copyright 2012 American Mathematical Society
  • 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
  • MathSciNet review: 2962179