Skip to main content
SpringerLink
Log in

Unitriangular factorizations of chevalley groups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU - UU - …U ± of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U - = U -(Φ, R) of the opposite Borel subgroup B - = B -(Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU - UU - U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU - UU - of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann’s Hypothesis, Chevalley groups over the ring \( \mathbb{Z}\left[ {\frac{1}{p}} \right] \) admit the unitriangular factorization G = UU - UU - UU - of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for SL n (n ≥ 3) and Sp2n (n ≥ 3), ” Publ. Math. Inst. Hautes Etudes Sci., 33, 59-137 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  2. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a commutative ring,” J. Soviet Math., 116, 1848-1860 (1984).

    Article  Google Scholar 

  3. N. A. Vavilov and S. S. Sinchuk, “Dennis-Vaserstein type decomposition, ” Zap. Nauchn. Semin. POMI, 375, 48-60 (2010).

    MathSciNet  Google Scholar 

  4. N. A. Vavilov and S. S. Sinchuk, “Parabolic factorizations of the split classical groups,” Algebra Analiz, 23, 48-60 (2011).

    MathSciNet  Google Scholar 

  5. A. Yu. Luzgarev and A. K. Stavrova, “Elementary subgroup of an isotropic reductive group is perfect,” Algebra Analiz, 23 (2011) (to appear).

  6. V. A. Petrov and A. K. Stavrova, "Elementary subgroups of isotropic reductive groups," St.Petersburg Math. J., 20, No. 3, 160-188 (2008).

    MathSciNet  Google Scholar 

  7. J.-P. Serre, "Le problème des groupes de congruence pour SL2,” Ann. Math., 92, 489-527 (1970).

    Article  MATH  Google Scholar 

  8. R. Steinberg, Lectures on Chevalley Groups, Yale University (1967).

  9. O. I. Tavgen, “Finite width of arithmetic subgroups of Chevalley groups of rank ≥ 2,” Soviet Math. Dokl., 41, No. 1, 136-140 (1990).

    MathSciNet  MATH  Google Scholar 

  10. O. I. Tavgen, “Bounded generation of Chevalley groups over rings of S-integer algebraic numbers,” Izu. Akad. Nauk USSR, 54, No. 1, 97-122 (1990).

    MATH  Google Scholar 

  11. E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J., 21. No. 3, 474-494 (1969).

    Article  MATH  Google Scholar 

  12. E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 1,185-198 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Babai, N. Nikolov, and L. Pyber, “Product growth and mixing in finite groups,” in: The 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM-SIAM (2008), pp. 248-257.

  14. H. Bass, “K-theory and stable algebra,” Publ. Math. Inst. Hautes Études Sci., No. 22, 5-60 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Carter and G. Keller, “Bounded elementary generation of SL n (\( \mathcal{O} \)),” Amer. J. Math., 105, 673-687 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Carter and G. Keller, “Elementary expressions for unimodular matrices,” Commun. Algebra, 12, 379-389 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Carter, G. E. Keller, and E. Paige, “Bounded expressions in SL(2, \( \mathcal{O} \)),” Preprint Univ. Virginia (1983).

    Google Scholar 

  18. R. VV. Carter, Simple Groups of Lie Type, Wiley, London et al. (1972).

    MATH  Google Scholar 

  19. Chen Baoquan and A. Kaufman, “3D volume rotation using shear transformations.” Graph. Models, 62, 308-322 (2000).

    Article  Google Scholar 

  20. Chen Huanyin and Chen Miaosen, “On products of three triangular matrices over associative rings,” Linear Algebra Appl., 387, 297-311 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Chernousov, E. Ellers, and N. Gordeev, “Gauss decomposition with prescribed semisimple part: short proof,” J. Algebra, 229, 314-332 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. M. Cohn, “On the structure of the GL2 of a ring,” Publ. Math. Inst. Hautes Études Sci., No. 30. 5-53 (1967).

  23. G. Cooke and P. J. Weinberger, “On the construction of division chains in algebraic number rings, with applications to SL2,” Commun. Algebra, 3, 481-524 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  24. R. K. Dennis and L. N. Vaserstein, “On a question of M. Newman on the number of commutators,” J. Algebra, 118, 150-161 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  25. E. Ellers and N. Gordeev, “On the conjectures of J. Thompson and O. Ore,” Trans. Amer. Math. Soc., 350, 3657-3671 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  26. I. V. Erovenko and A. S. Rapinchuk, “Bounded generation of some S-arithmetic orthogonal groups,” C. R. Acad. Sci., 333, No. 5, 395-398 (2001).

    MathSciNet  MATH  Google Scholar 

  27. F. J. Grunewald, J. Mennicke, and L. N. Vaserstein, “On the groups SL2(\( \mathbb{Z} \)[x]) and SL2(K[x, y]),” Israel J. Math., 86, Nos. 1-3, 157-193 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  28. R. M. Guralnick and G. Malle, “Products of conjugacy classes and fixed point spaces," arXiv: 1005 . 3756.

  29. Hao Pengwei, “Customizable triangular factorizations of matrices,” Linear Algebra Appl., 382, 135-154 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  30. W. van der Kallen, “SL3(\( \mathbb{C} \)[x]) does not have bounded word length,” Lect. Notes Math., 966, 357-361 (1982).

    Article  Google Scholar 

  31. T. J. Laffey, “Expressing unipotent matrices over rings as products of involutions,” Preprint Univ. Dublin (2010).

    Google Scholar 

  32. T. J. Laffey, Lectures on Integer Matrices, Beijing (2010).

  33. Lei Yang, Hao Pengwei, and Wu Dapeng, “Stabilization and optimization of PLUS factorization and its application to image coding,” J. Visual Comm. Image Repr., 22, No. 1 (2011).

  34. M. Larsen and A. Shalev, “Word maps and Waring type problems,” J. Amer. Math. Soc., 22, 437-466 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  35. H. W. Lenstra (jr.), P. Moree, and P. Stevenhagen, “Character sums for primitive root densities” (2011) (to appear).

  36. M. Liebeck and A. Shalev, “Classical groups, probabilistic methods, and the (2, 3)-generation problem,” Ann. Math., 144, No. 1, 77-125 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Liebeck and A. Shalev, “Diameters of finite simple groups: sharp bounds and applications,” Ann. Math., 154, 383-406 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Liebeek, N. Nikolov, and A. Shalev, “Groups of Lie type as products of SL2 subgroups,” J. Algebra, 326, 201-207 (2011).

    Article  MathSciNet  Google Scholar 

  39. M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “The Ore conjecture,” J. Europ. Math. Soc., 12, 939-1008 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “Products of squares in finite simple groups,” Proc. Amer. Math. Soc. (2011).

    Google Scholar 

  41. M. Liebeck and L. Pyber, “Finite linear groups and bounded generation,” Duke Math. J., 107, 159-171 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  42. B. Liehl, “Beschränkte Wortlänge in SL2.” Math. Z., 186, 509-524 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  43. H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup. (4), 2, 1-62 (1969).

    MATH  Google Scholar 

  44. P. Moree, “On primes in arithmetic progression having a prescribed primitive root,” J. Number Theory, 78, 85-98 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  45. P. Moree, “On primes in arithmetic progression having a prescribed primitive root. II,” Funet. Approx. Comment. Math., 39, 133-144 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  46. D. W. Morris, “Bounded generation of SL(n,A) (after D. Carter, G. Keller, and E. Paige),” New York J. Math., 13, 383-421 (2007).

    MathSciNet  MATH  Google Scholar 

  47. K. R. Nagarajan, M. P. Devaasahayam, and T. Soundararajan, "Products of three triangular matrices over commutative rings,” Linear Algebra Appl., 348, 1-6 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  48. N. Nikolov, “A product decomposition for the classical quasisimple groups,” J. Group Theory, 10, 43-53 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  49. N. Nikolov and L. Pyber, “Product decomposition of quasirandom groups and a Jordan type theorein,” arX:i.v:math/0703343 (2007).

  50. A. Paeth, “A fast algorithm for general raster rotation,” in: Graphics Gems, Acad. Press (1990), pp. 179-195.

  51. A. S. Rapinchuk and I. A. Rapinchuk, “Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank > 1 over Noetherian rings,” 1-12 (2010); arXiv:1007.2261v1 [math.GR].

  52. A. Shalev, “Cornrnutators, words, conjugaey classes, and character methods,” Turk. J. Math., 31, 131-148 (2007).

    MathSciNet  MATH  Google Scholar 

  53. A. Shalev, “Word maps, conjugacy classes, and a noncommutative Waring-type theorem,” Ann. Math., 170, No. 3, 1383-1416 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  54. R. W. Sharpe, “On the structure of the Steinberg group St(Λ),” J. Algebra, 68, 453-467 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  55. She Yiyuan and Hao Pengwei, “On the necessity and sufficiency of PLUS factorizations,” Linear Algebra Applic., 400, 193-202 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  56. S. Sinchuk and N. Vavilov, “Parabolic factorizations of exceptional Chevalley groups” (to appear).

  57. A. Sivatski and A. Stepanov, “On the word length of commutators in GL n (R),” K-theory, 17 (1999).

  58. M. R. Stein, “Surjective stability in dimension 0 f`or K2 and related functors,” Trans. Amer. Math. Soc., 178,176-191 (1973).

    Google Scholar 

  59. A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,” Israel Math. J., ??, 1-20 (2011).

  60. G. Strang, “Every unit matrix is a LULU,” Linear Algebra Applic., 265, 165-172 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  61. O. I. Tavgen, “Bounded generation of norrnal and twisted Chevalley groups over the rings of S-integers,” Contemp. Math., 131, No. 1, 409-421 (1992).

    Article  MathSciNet  Google Scholar 

  62. T. Toffoli, “Almost every unit matrix is a ULU,” Linear Algebra Applic., 259, 31-38 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Toffoli and J. Quick, “Three dimensional rotations by three shears,” Graphic. M orlels Image Process., 59, 89-96 (1997).

    Article  Google Scholar 

  64. L. N. Vaserstein, “Bass’ first stable range eondition,” J. Pure Appl. Algebra, 34, Nos. 2-3, 319-330 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  65. L. N. Vaserstein and E. Wheland, “Commutators and companion matrices over rings of stable rank 1,” Linear Algebra Appl., 142, 263-277 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  66. N. Vavilov, “Structure of Chevalley groups over commutative rings.” in: The Proceedings of the Conference on Nonassociative Algebras and Related Topics (Hiroshima-1990), World Sci. Publ., London et al. (1991), pp. 219-335.

    Google Scholar 

  67. N. Vavilov and E. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73-115 (1996).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    N. A. Vavilov & A. V. Smolensky

  2. Indian Statistics Institute Bangalore, Bangalore, India

    B. Sury

Authors
  1. N. A. Vavilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Smolensky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Sury
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Vavilov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 388, 2011, pp. 17-47.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Smolensky, A.V. & Sury, B. Unitriangular factorizations of chevalley groups. J Math Sci 183, 584–599 (2012). https://doi.org/10.1007/s10958-012-0826-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-0826-z

Keywords

Search

Navigation