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Journal of Mathematical Sciences

, Volume 219, Issue 3, pp 355–369 | Cite as

Decomposition of Unipotents for E6 and E7: 25 Years After

  • N. A. VavilovEmail author
Article

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations (E61) and (E77). To put them in the context, I first very briefly recall the two previous stages of the method, an A5-proof for E6 and an A7-proof for E7, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin, and myself (a definitive exposition was given in my paper “A third look at weight diagrams”) and an A2-proof for E6 and E7 developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the A2-proof actually effectuates reduction to small parabolics, of corank 3 in E6 and corank 5 in E7. This allows one to revamp proofs and to sharpen existing bounds in many applications. The second new variation is a D5-proof for E6, based on stabilization of columns with one zero. [I devised also a similar D6-proof for E7, based on stabilization of columns with two adjacent zeros, but it is too abstruse to be included in a casual exposition.] Also, I list several further variations. Actual detailed calculations will appear in my paper “A closer look at weight diagrams of types (E61) and (E77).”

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References

  1. 1.
    A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq., 7, No. 2, 159–196 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Hazrat, A. Stepanov, N. Vavilov, and Zuhong Zhang, Commutators width in Chevalley groups,” Note di Matematica, 33, No. 1, 139–170 (2013).Google Scholar
  3. 3.
    R. Hazrat and N. Vavilov, “K 1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, No. 1, 99–116 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    R. Hazrat and N. Vavilov, “Bak’s work on the K-theory of rings,” J. K-Theory, 4, No. 1, 1–65 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. I. Kopeiko, “Stabilization of symplectic groups over a ring of polynomials,” Math. USSR Sb., 34, No. 5, 655–669 (1978).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Yu. Luzgarev and A. K. Stavrova, “The elementary subgroup of an isotropic reductive group is perfect,” St. Petersburg Math. J., 23, No. 5, 881–890 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup., (4) 2, 1–62 (1969).Google Scholar
  8. 8.
    V. A. Petrov and A. K. Stavrova, “Elementary subgroups in isotropic reductive groups,” St. Petersburg Math. J., 20, No. 4, 625–644 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. Petrov and A. Stavrova, “The Tits indices over semilocal rings,” Transform. Groups, 16, No. 1, 193–217 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. Plotkin, A. Semenov, and N. Vavilov, “Visual basic representations: an atlas,” Internat. J. Algebra Comput., 8, No. 1, 61–95 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Sivatsky and A. Stepanov, “On the word length of commutators in GLn(R),” K-theory, 17, 295–302 (1999).MathSciNetCrossRefGoogle Scholar
  12. 12.
    M. R. Stein, “Stability theorems for K 1, K 2 and related functors modeled on Chevalley groups,” Japan. J. Math. (N.S.), 4, No. 1, 77–108 (1978).Google Scholar
  13. 13.
    A. Stepanov, “Stability conditions in the theory of linear groups over rings,” Ph. D. Thesis, Leningrad State Univ. (1987).Google Scholar
  14. 14.
    A. Stepanov, “Structure of Chevalley groups over rings via universal localization,” J. KTheory, 1–18 (2014), in press.; see arXiv:1303.6082v3 [math.RA].Google Scholar
  15. 15.
    A. Stepanov, “Non-abelian K-theory of Chevalley groups over rings,” J. Math. Sci. (N. Y.) (2014), in press.Google Scholar
  16. 16.
    A. V. Stepanov, “Structure theory and subgroups of Chevalley groups over rings,” Habilitationsschrift Saint Petersburg State Univ. (2014).Google Scholar
  17. 17.
    A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, No. 2, 109–153 (2000).Google Scholar
  18. 18.
    A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,” Israel J. Math., 185, No. 1, 253–276 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. A. Suslin, “The structure of the special linear group over polynomial rings,” Math. USSR Izv., 11, No. 2, 235–253 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. A. Suslin and V. I. Kopeiko, “Quadratic modules and the orthogonal group over polynomial rings,” J. Soviet Math., 20, No. 6, 2665–2691 (1982).CrossRefzbMATHGoogle Scholar
  21. 21.
    G. Taddei, “Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, 693–710 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    N. A. Vavilov, “Subgroups of split classical groups,” Habilitationsschrift Leningrad State Univ. (1987).Google Scholar
  23. 23.
    N. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, NJ (1991), pp. 219–335.Google Scholar
  24. 24.
    N. Vavilov, “A third look at weight diagrams,” Rend. Sem. Mat. Univ. Padova, 104, 201–250 (2000).MathSciNetzbMATHGoogle Scholar
  25. 25.
    N. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7,” Internat. J. Algebra Comput., 17, No. 5-6, 1283–1298 (2007).Google Scholar
  26. 26.
    N. A. Vavilov, “Can one see the signs of structure constants?” St. Petersburg Math. J., 19, No. 4, 519–543 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    N. A. Vavilov, “Numerology of quadratic equations,” St. Petersburg Math. J., 20, No. 5, 687–707 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    N. A. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7. II. Fundamental lemma”, St. Petersburg Math. J., 23, No. 6, 921–942 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    N. Vavilov, “A closer look at weight diagrams of types (E6 , \( \varpi \) 1) and (E7 , \( \varpi \) 7),” 1–48, to appear (2014).Google Scholar
  30. 30.
    N. A. Vavilov and M. R. Gavrilovich, “A2-proof of structure theorems for Chevalley groups of types E6 and E7,” St. Petersburg Math. J., 16, No. 4, 649–672 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book,” J. Math. Sci., 140, No. 5, 626–645 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    N. A. Vavilov and V. G. Kazakevich, “Yet another variation on the theme of decomposition of transvections,” Vestnik St. Petersburg Univ. Math., 41, No. 4, 345–347 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    N. A. Vavilov and V. G. Kazakevich, “More variations on the decomposition of transvections,” J. Math. Sci., 171, No. 3, 322–330 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E6,” St. Petersburg Math. J., 19, No. 5, 699–718 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    N. A. Vavilov and A. Yu. Luzgarev, “Chevalley group of type E7 in the 56-dimensional representation,” J. Math. Sci., 180, No. 3, 197–251 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    N. A. Vavilov and A. Yu. Luzgarev, “A2-proof of structure theorems for the Chevalley group of type E8,” St. Petersburg Math. J. (2014), to appear.Google Scholar
  37. 37.
    N. A. Vavilov and A. Yu. Luzgarev, “Calculations in exceptional groups, an update,” J. Math. Sci., 1–13 (2014), in press.Google Scholar
  38. 38.
    N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “Chevalley group of type E6 in theGoogle Scholar
  39. 39.
    27-dimensional representation,” J. Math. Sci., 145, No. 1, 4697–4736 (2007).Google Scholar
  40. 40.
    N. Vavilov, A. Luzgarev, and A. Stepanov, “Calculations in exceptional groups over rings,” J. Math. Sci., 373, 48–72 (2009).MathSciNetzbMATHGoogle Scholar
  41. 41.
    N. A. Vavilov and S. I. Nikolenko, “A2-proof of structure theorems for the Chevalley group of type F4,” St. Petersburg Math. J., 20, No. 4, 527–551 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    N. A. Vavilov and E. Ya. Perelman, “Polyvector representations of GLn,” J. Math. Sci., 145, No. 1, 4737–4750 (2007).MathSciNetCrossRefGoogle Scholar
  43. 43.
    N. A. Vavilov and E. B. Plotkin, “Stabilization of columns,” unpublished manuscript, 1–12 (1989).Google Scholar
  44. 44.
    N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, No. 1, 73–113 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Calculations in Chevalley groups over commutative rings,” Soviet Math. Dokl., 40, No. 1, 145–147 (1990).MathSciNetzbMATHGoogle Scholar
  46. 46.
    N. A. Vavilov and A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels,” J. Math. Sci., 192, No. 2, 164–195 (2013).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt. PetersburgRussia

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