Skip to main content
SpringerLink
Log in

Essential p-dimension of the normalizer of a maximal torus

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We compute the exact value of the essential p-dimension of the normalizer of a split maximal torus for most simple connected linear algebraic groups. These values give new upper bounds on the essential p-dimension of some simple groups, including some exceptional groups. For each connected simple algebraic group, we also give an upper bound on the essential p-dimension of any torus contained in that group. These results are achieved by a detailed case-by-case analysis.

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. G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math. 8 (2003), 279–330.

    MathSciNet  MATH  Google Scholar 

  2. A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  3. N. Bourbaki, Groupes et Algèbres de Lie, Chaps. IV–VI, Hermann, Paris, 1968. Russian transl.: Н. Бурбаки, Группы и алгебры Ли, главы IV–VI, Мир, M., 1972.

    MATH  Google Scholar 

  4. J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159–179.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Chernousov, Another proof of Totaro's theorem on E 8 -torsors, Canad. Math. Bull. 49 (2006), no. 2, 196–202.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Chernousov, A. S. Merkurjev, Essential p-dimension of split simple groups of type A n , preprint: http://www.mathematik.uni-bielefeld.de/LAG/man/429.html.

  7. J. H. Conway, N. J. A. Sloane, Low-dimensional lattices, I. Quadratic forms of small determinant, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 17–41.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. H. Conway, N. J. A. Sloane, Low-dimensional lattices, II. Subgroups of \( {\text{GL}}\left( {n,\mathbb{Z}} \right) \), Proc. Roy. Soc. London Ser. A 419 (1988), no. 1856, 29–68.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, New York (1993). Russian transl.: Дж. Конвей, Н. Слоэн, Упаковки щаров, рещетки и группы, тт. I, II, Мир, M., 1990.

    MATH  Google Scholar 

  10. A. Duncan, Essential dimensions of A 7 and S 7, Math. Res. Lett. 17 (2010), no. 2, 263–266.

    MathSciNet  MATH  Google Scholar 

  11. G. Favi, M. Florence, Tori and essential dimension, J. Algebra 319 (2008), no. 9, 3885–3900.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Gille, Type des tores maximaux des groupes semi-simples, J. Ramanujan Math. Soc. 19 (2004), no. 3, 213–230.

    MathSciNet  MATH  Google Scholar 

  13. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29. Cambridge University Press, Cambridge, 1992.

    MATH  Google Scholar 

  14. J.C. Jantzen, Representations of Algebraic Groups, Pure and Applied Mathematics, Vol. 131. Academic Press, Inc., Boston, MA, 1987.

    MATH  Google Scholar 

  15. N. A. Karpenko, A. S. Merkurjev, Essential dimension of finite p-groups, Invent. Math. 172 (2008), no. 3, 491–508.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Lemire, Essential dimension of algebraic groups and integral representations of Weyl groups, Transform. Groups 9 (2004), no. 4, 337–379.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Lötscher, M. L. MacDonald, A. Meyer, Z. Reichstein, Essential p-dimension of algebraic tori, preprint: http://www.math.uni-bielefeld.de/LAG/man/363.html.

  18. R. Lötscher, M. L. MacDonald, A. Meyer, Z. Reichstein, Essential dimension of algebraic tori, preprint: http://www.math.uni-bielefeld.de/LAG/man/399.html.

  19. M.L. MacDonald, Cohomological invariants of odd degree Jordan algebras, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 2, 295–303.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra Number Theory 3 (2009), no. 4, 467–487.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Meyer, Z. Reichstein, Some Consequences of the Karpenko-Merkurjev theorem, Documenta Math. Extra Volume: Andrei A. Suslin’s Sixtieth Birthday (2010), 445–457.

  22. M. S. Raghunathan, Tori in quasi-split-groups, J. Ramanujan Math. Soc. 19 (2004), no. 4, 281–287.

    MathSciNet  MATH  Google Scholar 

  23. Z. Reichstein, Essential dimension, Proceedings of ICM 2010, to appear, preprint: http://www.mathematik.uni-bielefeld.de/LAG/man/393.html.

  24. Z. Reichstein, B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, with an appendix by J. Kollár and E. Szabó. Canad. J. Math. 52 (2000), no. 5, 1018–1056.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.-P. Serre, Galois Cohomology, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  26. J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Vol. IX, Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62.

    Google Scholar 

  27. В. Е. Воскресенский, Максималные торы без аффекта в полупростых алгебраических группах, Мат. заметки 44 (1988), no. 3, 309–318, 410. Engl. transl.: V. E. Voskresenskii, Maximal tori without affect in semisimple algebraic groups, Math. Notes 44 (1988), nos. 3–4, 651–655 (1989).

    MathSciNet  Google Scholar 

  28. V. E. Voskresenskii, Algebraic Groups and Their Birational Invariants, Translations of Mathematical Monographs, Vol. 179, American Mathematical Society, Providence, RI, 1998.

    Google Scholar 

  29. W. C. Waterhouse, Introduction of Affine Group Schemes, Graduate Texts in Mathematics, Vol. 66. Springer-Verlag, New York, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Mark L. MacDonald

Authors
  1. Mark L. MacDonald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark L. MacDonald.

Additional information

Partially supported by an NSERC Postdoctoral Fellowship. (Mark L. MacDonald)

Rights and permissions

Reprints and permissions

About this article

Cite this article

MacDonald, M.L. Essential p-dimension of the normalizer of a maximal torus. Transformation Groups 16, 1143–1171 (2011). https://doi.org/10.1007/s00031-011-9157-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-011-9157-2

Keywords

Search

Navigation