Skip to main content
Log in

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript


We describe the null-cone of the representation of G on M p, where either G  =  SL(W)  ×  SL(V) and M  =  Hom(V,W) (linear maps), or G  =  SL(V) and M is one of the representations S 2(V *) (symmetric bilinear forms), Λ2(V *) (skew bilinear forms), or \(V^* \otimes V^*\) (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M p. Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M p is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. Adamovich O., Golovina E. (1977). Invariants of a pair of bilinear forms. Mosc. Univ. Math. Bull. 32(2):11–14

    MATH  Google Scholar 

  2. Bollobás B. (1998). Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  3. Buhler J., Gupta R., Harris J. (1987). Isotropic subspaces of skewforms and maximal abelian subgroups of p-groups. J. Algebra 108:269–279

    Article  MATH  MathSciNet  Google Scholar 

  4. Draisma, J.: Counting components of the null-cone on tuples. Preprint (2005). Presented at MEGA 2005, to appear in Transformation Groups.

  5. Draisma, J.: Small maximal spaces of non-invertible matrices. Bull. Lond. Math. Soc. (Preprint, 2005, to appear)

  6. Draisma, J., Kemper, G., Wehlau, D.: Polarization of separating invariants (Submitted, 2005)

  7. Eisenbud D., Harris J. (1988). Vector spaces of matrices of low rank. Adv. Math. 70(2):135–155

    Article  MATH  MathSciNet  Google Scholar 

  8. Gantmacher F. (1959). The theory of matrices, vol. 2. AMS Chelsea Publishing, New York

    MATH  Google Scholar 

  9. Goodman R., Wallach N.R. (1998). Representations and Invariants of the Classical Groups. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  10. Harris, J.: Algebraic geometry. A first course. No. 133 in Graduate Texts in Mathematics. Springer, Berlin Heidelberg New York (1992)

  11. Hesselink W.H. (1979). Desingularizations of varieties of nullforms. Invent. Math. 55:141–163

    Article  MATH  MathSciNet  Google Scholar 

  12. Knop, F.: On Noethers and Weyl’s bound in positive characteristic. In: Invariant theory in all Characteristics. CRM Proceedings of Lecture Notes, vol. 35, pp. 175–188. Am. Math. Soc., Providence, RI (2004)

  13. Kraft H. (1984). Geometrische Methoden in der Invariantentheorie. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden

    MATH  Google Scholar 

  14. Kraft, H., Wallach, N.R.: On the nullcone of representations of reductive groups. Pac. J. Math. (Preprint, 2005).

  15. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 34. Springer, Berlin Heidelberg New York (1993)

  16. Popov, V., Vinberg, E.: Invariant Theory, Encyclopaedia of Mathematical Sciences, vol. 55, chap. II. Springer, Berlin Heidelberg New York (1994)

  17. Wall C. (1978). Nets of quadrics, and theta-characteristics of singular curves. Philos. Trans. Roy. Soc. London Ser. A 289(1357):229–269

    MATH  MathSciNet  Google Scholar 

  18. Weyl H. (1939). The classical groups, their invariants and representations. Princeton University Press, Princeton

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matthias Bürgin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bürgin, M., Draisma, J. The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms. Math. Z. 254, 785–809 (2006).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification (2000)