Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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
DOI: https://doi.org/10.1090/S1061-0022-2012-01223-9
Published electronically: September 17, 2012
MathSciNet review: 2962179
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. E. Abe, Automorphisms of Chevalley groups over commutative rings, Algebra i Analiz 5 (1993), no. 2, 74-90; English transl., St. Petersburg Math. J. 5 (1994), no. 2, 287-300. MR 1223171 (94e:20062)
  • 2. A. Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (Inst. Adv. Study, Princeton, NJ, 1968/1969), Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin-New York, 1970, pp. 1-55. MR 0258838 (41:3484)
  • 3. N. Bourbaki, Lie groups and Lie algebras. Ch. 4-6, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
  • 4. N. A. Vavilov, Can one see the signs of structure constants? Algebra i Analiz 19 (2007), no. 4, 34-68; English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519-543. MR 2381932 (2009b:20087)
  • 5. -, Numerology of quadratic equations, Algebra i Analiz 20 (2008), no. 5, 9-40; English transl., St. Petersburg Math. J. 20 (2009), no. 5, 687-707. MR 2492358 (2010a:20094)
  • 6. N. A. Vavilov and M. R. Gavrilovich, An $ A_2$-proof of structure theorems for Chevalley groups of types $ \mathrm E_6$ and $ \mathrm E_7$, Algebra i Analiz 16 (2004), no. 4, 54-87; English transl., St. Petersburg Math. J. 16 (2005), no. 4, 649-672. MR 2090851 (2005m:20115)
  • 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), 36-76; English transl., J. Math. Sci. (N. Y.) 140 (2007), no. 5, 626-645. MR 2253566 (2007k:20108)
  • 8. N. A. Vavilov and V. G. Kazakevich, Yet another variation on the theme of decomposition of transvections, Vestnik S.-Peterburg. Univ. Ser. 1 2008, vyp. 4, 71-74; English transl., Vestnik St. Petersburg Univ. Math. 41 (2008), no. 4, 345-347. MR 2485399
  • 9. -, Decomposition of transvections for automorphisms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 365 (2009), 47-62; English transl., J. Math. Sci (N. Y.) 161 (2009), no. 4, 483-491. MR 2749134
  • 10. -, More variations on the decomposition of transvections, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 375 (2010), 32-47; English transl., J. Math. Sci. (N. Y.) 171 (2010), no. 3, 322-330. MR 2749273 (2012a:20081)
  • 11. N. A. Vavilov and A. Yu. Luzgarev, The normalizer of Chevalley groups of type $ \mathrm E_6$, Algebra i Analiz 19 (2007), no. 5, 37-64; English transl., St. Petersburg Math. J. 19 (2008), no. 5, 699-718. MR 2381940 (2008m:20077)
  • 12. -, The normalizer of Chevalley groups of type $ \mathrm E_7$ (to appear). (Russian)
  • 13. -, Chevalley groups of type $ \mathrm E_7$ in the $ 56$-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), 5-99. (Russian) MR 2784131
  • 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, Chevalley groups of type $ \mathrm E_6$ in the $ 27$-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), 5-68; English transl., J. Math. Sci. (N. Y.) 145 (2007), no. 1, 4697-4736. MR 2354606 (2009b:20022)
  • 16. N. A. Vavilov and S. I. Nikolenko, $ \mathrm A_2$-proof of structure theorems for Chevalley groups of type $ \mathrm F_4$, Algebra i Analiz 20 (2008), no. 4, 27-63; English transl., St. Petersburg Math. J. 20 (2009), no. 4, 527-551. MR 2473743 (2010a:20098)
  • 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; English transl., Soviet Math. Dokl. 40 (1990), no. 1, 145-147. MR 1020667 (90j:20093)
  • 18. N. A. Vavilov and A. K. Stavrova, Basic reduction in the problem of the description of normal subgroups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), 30-52; English transl., J. Math. Sci. (N. Y.) 151 (2008), no. 3, 2949-2960. MR 2742853 (2011k:20097)
  • 19. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. (N. S.) 30 (1952), no. 2, 349-362. (Russian) MR 0047629 (13:904c)
  • 20. A. Yu. Luzgarev, On overgroups $ E(\mathrm E_6,R)$ and $ E(\mathrm E_7,R)$ in minimal representations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), 216-243; English transl., J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2558-2571. MR 2117858 (2006k:20098)
  • 21. -, Description of the overgroups $ \mathrm F_4$ in $ \mathrm E_6$ over a commutative ring, Algebra i Analiz 20 (2008), no. 6, 148-185; English transl., St.-Petersburg Math. J. 20 (2009), no. 6, 955-981. MR 2530897 (2010f:20045)
  • 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 $ \mathrm E_6$, Algebra i Analiz 23 (2011), no. 3, 261-309; English transl. in St. Petersburg Math. J. 23 (2012), no. 3. MR 2896171
  • 24. -, Width of groups of type $ \mathrm E_6$ with respect to root elements. I, Algebra i Analiz 23 (2011), no. 5, 155-198; English transl. in St. Petersburg Math. J. 23 (2012), no. 5. MR 2896171
  • 25. T. A. Springer, Linear algebraic groups, 2nd ed., Progr. in Math., vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713 (99h:20075)
  • 26. R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, 1968. MR 0466335 (57:6215)
  • 27. M. S. Tulenbaev, The Schur multiplier of the group of elementary matrices of finite order, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 86 (1979), 162-169; English transl. in J. Soviet Math. 17 (1981), no. 4. MR 0535488 (80j:18019)
  • 28. E. Abe, Chevalley groups over local rings, Tôhoku Math. J. (2) 21 (1969), no. 3, 474-494. MR 0258837 (41:3483)
  • 29. -, Chevalley groups over commutative rings, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, pp. 1-23. MR 0999577 (91a:20047)
  • 30. -, 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 0991973 (91a:20046)
  • 31. E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185-198. MR 0439947 (55:12828)
  • 32. R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), no. 1, 1-59. MR 0318337 (47:6884)
  • 33. -, Simple groups of Lie type, Pure Appl. Math., vol. 28, Wiley, London, 1972. MR 0407163 (53:10946)
  • 34. W. van der Kallen, Another presentation for Steinberg groups, Nederl. Akad. Wetensch. Proc. Ser. A 80 (1977), 304-312. MR 0463263 (57:3216)
  • 35. H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1-62. MR 0240214 (39:1566)
  • 36. V. Petrov and A. Stavrova, Tits indices over semilocal rings, Transform. Groups 16 (2011), no. 1, 193-217. MR 2785501 (2012c:20144)
  • 37. E. Plotkin, On the stability of the $ K_1$-functor for Chevalley groups of type $ \mathrm E_7$, J. Algebra 210 (1998), 67-85. MR 1656415 (99k:20099)
  • 38. E. Plotkin, A. Semenov, and N. Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61-95. MR 1492062 (98m:17010)
  • 39. M. R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), no. 4, 965-1004. MR 0322073 (48:437)
  • 40. -, 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 0528869 (81c:20031)
  • 41. A. Stepanov and N. Vavilov, Decomposition of transvections: a theme with variations, $ K$-Theory 19 (2000), 109-153. MR 1740757 (2000m:20076)
  • 42. N. 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 (92k:20090)
  • 43. -, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201-250. MR 1809357 (2001i:20099)
  • 44. -, Do it yourself structure constants for Lie algebras of type $ \mathrm E_l$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), 60-104; English transl., J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1513-1548. MR 1875718 (2002k:17022)
  • 45. -, An $ \mathrm A_3$-proof of structure theorems for Chevalley groups of types $ \mathrm E_6$ and $ \mathrm E_7$, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283-1298. MR 2355697 (2009h:20054)
  • 46. N. A. Vavilov, A. Yu. Luzgarev, and A. V. Stepanov, Calculations in exceptional groups over rings, Zap. Nauchn. Sem. S.-Peterburg Otdel. Mat. Inst. Steklov. (POMI) 373 (2009), 48-72; English transl., J. Math. Sci. (N. Y.) 168 (2010), no. 3, 334-348. MR 2749253 (2012c:20136)
  • 47. N. Vavilov and E. Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), 73-115. MR 1409655 (97h:20056)
  • 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).

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 20G15, 20G35

Retrieve articles in all journals with MSC (2010): 20G15, 20G35


Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: nikolai-vavilov@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01223-9
Keywords: Chevalley groups, elementary subgroups, normal subgroups, standard description, minimal modules, parabolic subgroups, decomposition of unipotents, root elements, orbits of highest weight vector, the Proof from the Book
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.
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society