Archiv der Mathematik

, Volume 107, Issue 2, pp 101–119 | Cite as

Essential dimension of algebraic groups, including bad characteristic

  • Skip Garibaldi
  • Robert M. Guralnick


We give upper bounds on the essential dimension of (quasi-) simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group G of rank at least two is at most dim G - 2(rank G) - 1. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good as or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.

Mathematics Subject Classification

Primary 11E72 Secondary 20G41 17B45 


Essential dimension Adjoint representation Generic stabilizer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Auel A., Brussel E., Garibaldi S., Vishne U.: Open problems on central simple algebras. Transform. Groups 16, 219–264 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andreev E.M., Popov V.L.: Stationary subgroups of points of general position in the representation space of a semisimple Lie group. Funct. Anal. Appl. 5, 265–271 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baek S.: Essential dimension of simple algebras in positive characteristic. C. R. Acad. Sci. Paris Sér. I Math. 349, 375–378 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baek S., Merkurjev A.: Essential dimension of central simple algebras. Acta Math. 209, 1–27 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Babic A., Chernousov V.: Lower bounds for essential dimensions in characteristic 2 via orthogonal representations. Pacific J. Math. 279, 37–63 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    N. Bourbaki, Lie groups and Lie algebras: Chapters 4–6, Springer-Verlag, Berlin, 2002.Google Scholar
  7. 7.
    T. Burness, R. Guralnick, and J. Saxl, On base sizes for algebraic groups, arXiv:1310.1569, to appear in J. Eur. Math. Soc.
  8. 8.
    P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension and algebraic stacks, arXiv:math/0701903 , 2007.
  9. 9.
    P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544.Google Scholar
  10. 10.
    Chernousov V., Gille P., Reichstein Z.: Reduction of structure for torsors over semilocal rings. Manuscripta Math. 126, 465–480 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chernousov V., Merkurjev A.: Essential p-dimension of split simple groups of type A n. Math. Ann. 357, 1–10 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chernousov V., Serre J-P.: Lower bounds for essential dimensions via orthogonal representations. J. Algebra 305, 1055–1070 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Demazure and A. Grothendieck, Schémas en groupes II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, vol. 152, Springer, 1970.Google Scholar
  14. 14.
    Duncan A., Reichstein Z.: Versality of algebraic group actions and rational points on twisted varieties. J. Algebraic Geom. 24, 499–530 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. Garibaldi, Cohomological invariants: exceptional groups and spin groups, Memoirs Amer. Math. Soc., no. 937, Amer. Math. Soc., Providence, RI, 2009, with an appendix by Detlev W. Hoffmann.Google Scholar
  16. 16.
    S. Garibaldi and R.M. Guralnick, Simple groups stabilizing polynomials, Forum of Math. Pi 3 (2015), e3 (41 pages).Google Scholar
  17. 17.
    S. Garibaldi and R.M. Guralnick, Spinors and essential dimension, preprint, 2016, arXiv:1601.00590.
  18. 18.
    Gille P., Reichstein Z.: A lower bound on the essential dimension of a connected linear group. Comment. Math. Helv. 84, 189–212 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge studies in Advanced Math., vol. 101, Cambridge, 2006.Google Scholar
  20. 20.
    Goldstein D., Guralnick R.: Alternating forms and self-adjoint operators. J. Algebra 308, 330–349 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    R.M. Guralnick, R. Lawther, M. Liebeck, and D. Testerman, Generic stabilizers for actions of simple algebraic groups, in preparation, 2016.Google Scholar
  22. 22.
    J.C. Jantzen, Representations of algebraic groups, second ed., Math. Surveys and Monographs, vol. 107, Amer. Math. Soc., Providence, RI, 2003.Google Scholar
  23. 23.
    M.-A. Knus, A.S. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, Colloquium Publications, vol. 44, Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  24. 24.
    Lemire N.: Essential dimension of algebraic groups and integral representations of Weyl groups. Transformation Groups 9, 337–379 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    R. Lötscher and M. MacDonald, The slice method for G-torsors, preprint, 2015.Google Scholar
  26. 26.
    R. Lötscher, M. MacDonald, A. Meyer, and Z. Reichstein, Essential dimension of algebraic tori, J. Reine Angew. Math. 677 (2013), 1–13.Google Scholar
  27. 27.
    M. Lorenz, Z. Reichstein, L. Rowen, and D. Saltman, Fields of definition for division algebras, J. London Math. Soc. (2) 68 (2003), 651–670.Google Scholar
  28. 28.
    MacDonald M.L.: Cohomological invariants of odd degree Jordan algebras. Math. Proc. Cambridge Philos. Soc. 145, 295–303 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    MacDonald M.L.: Upper bounds for the essential dimension of E 7. Canad Math. Bull. 56, 795–800 (2013)MathSciNetzbMATHGoogle Scholar
  30. 30.
    MacDonald M.L.: Essential dimension of Albert algebras. Bull. Lond. Math. Soc. 46, 906–914 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Merkurjev A.: A lower bound on the essential dimension of simple algebras. Algebra & Number Theory 4, 1055–1076 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Merkurjev A.: Essential dimension: a survey. Transform. Groups 18, 415–481 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Merkurjev, Essential dimension, Séminaire Bourbaki, 67ème année, #1101, June 2015.Google Scholar
  34. 34.
    Meyer A., Reichstein Z.: The essential dimension of a maximal torus in the projective linear group. Algebra & Number Theory 3, 467–487 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zur H.: Theorie der positiven quadratischen Formen J. Reine Angew. Math. 101, 196–202 (1887)MathSciNetGoogle Scholar
  36. 36.
    A.M. Popov, Finite isotropy subgroups in general position of simple linear Lie groups, Trans. Moscow Math. Soc. (1988), 205–249, [Russian original: Trudy Moskov. Mat. Obschch. 50 (1987), 209–248, 262].Google Scholar
  37. 37.
    Reichstein Z.: On the notion of essential dimension for algebraic groups. Transform. Groups 5, 265–304 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Z. Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians, Volume II, Hindustan Book Agency, New Delhi, 2010.Google Scholar
  39. 39.
    R.W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J. (1988), 1–35.Google Scholar
  40. 40.
    M. Rost, On 14-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, unpublished note, March 1999.Google Scholar
  41. 41.
    M. Rost, On the Galois cohomology of Spin(14), unpublished note, March 1999.Google Scholar
  42. 42.
    J-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 2002, originally published as: Cohomologie galoisienne, 1965.Google Scholar
  43. 43.
    J-P. Serre, Bounds for the orders of the finite subgroups of G(k), Group representation theory, EPFL Press, Lausanne, 2007, pp. 405–450.Google Scholar
  44. 44.
    R. Steinberg, Torsion in reductive groups, Adv. Math. 15 (1975), 63–92, [= Collected Papers, pp. 415–444].Google Scholar
  45. 45.
    W.C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer, New York-Berlin, 1979.Google Scholar

Copyright information

© Springer International Publishing (outside the USA) 2016

Authors and Affiliations

  1. 1.Center for Communications ResearchSan DiegoUSA
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations