Algebras and Representation Theory

, Volume 17, Issue 6, pp 1683–1706 | Cite as

Non-Reductive Conjugation on the Nilpotent Cone

  • Magdalena BoosEmail author


We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL n (C), especially of the Borel subgroup B and of the standard unipotent subgroup U of the latter on the nilpotent cone of complex nilpotent matrices. We obtain generic normal forms of the orbits and describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the U-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related. The study of a GIT-quotient for the Borel-action is initiated.


Parabolic orbits in the nilpotent cone Semiinvariants Generic normal form (Semi-)invariant ring 

Mathematics Subject Classifications (2010)

14R20 16W22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575–611 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Boos, M.: Conjugation on varieties of nilpotent matrices. PhD thesis. Bergische Universität, Wuppertal (2012)Google Scholar
  3. 3.
    Boos, M., Reineke, M.: B-orbits of 2-nilpotent matrices. In: Highlights in Lie Algebraic Methods (Progress in Mathematics), pp. 147–166, Birkhäuser (2011)Google Scholar
  4. 4.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  5. 5.
    Fulton, W.: Introduction to toric varieties. In: Annals of Mathematics Studies, The William H. Roever Lectures in Geometry, vol. 131. Princeton University Press, Princeton (1993)Google Scholar
  6. 6.
    Halbach, B.: B-Orbiten nilpotenter Matrizen. Bachelorarbeit, Bergische Universität Wuppertal (2009)Google Scholar
  7. 7.
    Hilbert, D.: Ueber die Theorie der algebraischen Formen. Math. Ann. 36(4), 473–534 (1890)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kraft, H.: Geometrische Methoden in der Invariantentheorie, Vol. D1. Friedr. Vieweg & Sohn, Braunschweig (1984)CrossRefGoogle Scholar
  9. 9.
    Mukai, S.: An introduction to invariants and moduli. In: Studies in Advanced Mathematics, vol. 81. Cambridge University Press, Cambridge. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury (2003)Google Scholar
  10. 10.
    Nagata, M.: On the fourteenth problem of Hilbert. In: Proceedings on the International Congress Math. 1958, pp. 459–462. Cambridge University Press, New York (1960)Google Scholar
  11. 11.
    Reineke, M.: Moduli of representations of quivers. Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., 589–637. Europe Mathmetics Society, Zürich (2008)Google Scholar
  12. 12.
    Schofield, A., van den Bergh, M.: Semi-invariants of quivers for arbitrary dimension vectors. Indag. Math. (N.S.) 12(1), 125–138 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Serre, J.-P.: Espaces fibrés algébriques (d’après André Weil). In: Séminaire Bourbaki, vol. 2, 82, pp. 305–311. Society Mathematics France, Paris (1995)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Fachbereich C - MathematikBergische Universität WuppertalWuppertalGermany

Personalised recommendations