Advertisement

Transformation Groups

, Volume 24, Issue 4, pp 951–986 | Cite as

PARABOLIC CONJUGATION AND COMMUTING VARIETIES

  • M. BOOS
  • M. BULOISEmail author
Article
  • 22 Downloads

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Aitken, H. W. Turnbull, An Introduction to the Theory of Canonical Matrices, Dover, New York, 1961.zbMATHGoogle Scholar
  2. [2]
    I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1, Techniques of Representation Theory, London Mathematical Society Student, Vol. 65, Cambridge University Press, Cambridge, 2006.Google Scholar
  3. [3]
    K. Bongartz, P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Boos, Finite parabolic conjugation on varieties of nilpotent matrices, Algebr. Represent. Theory 17 (2014), no. 6, 1657–1682.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Boos, Staircase algebras and graded nilpotent pairs, J. Pure Appl. Algebra 221 (2017), no. 8, 2032–2052.MathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Brüstle, L. Hille, Finite, tame, and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups. J. Algebra 226 (2000), 347–380.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Bulois, L. Evain, Nested punctual hilbert schemes and commuting varieties of parabolic subalgebras, J. Lie Theory 26 (2016), no. 2, 497–533.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39–90.MathSciNetCrossRefGoogle Scholar
  9. [9]
    V. Dlab, C. M. Ringel, The module theoretical approach to quasi-hereditary algebras. in: Representations of Algebras and Related Topics, London Math. Soc. Lecture Note Ser., Vol. 168, Cambridge University Press, Cambridge, 1992, pp. 200–224.Google Scholar
  10. [10]
    P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103, correction, ibid., 309.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Gabriel, The universal cover of a representation-finite algebra, in: Representations of Algebras (Puebla, 1980), Lecture Notes in Math., Vol. 903, Springer, Berlin, 1981, pp. 68–105.Google Scholar
  12. [12]
    C. Geiss, B. Leclerc, J. Schröer, Quivers with relations for symmetrizable cartan matrices i: Foundations, Invent. Math. 209 (2017), no. 1, 61–158.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Goddard, S. M. Goodwin, On commuting varieties of parabolic subalgebras, J. Pure Appl. Algebra 222 (2018), no. 3, 481–507.MathSciNetCrossRefGoogle Scholar
  14. [14]
    S. M. Goodwin, G. Röhrle, On commuting varieties of nilradicals of borel subalgebras of reductive lie algebras, Proc. Edinburgh Math. Soc. (2) 58 (2015), 169–181.MathSciNetCrossRefGoogle Scholar
  15. [15]
    D. Happel, D. Vossieck, Minimal algebras of infinite representation type with pre-projective component, Manuscripta Math. 42 (1983), no. 2-3, 221–243.MathSciNetCrossRefGoogle Scholar
  16. [16]
    L. Hille, G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1999), no. 1, 35–52.MathSciNetCrossRefGoogle Scholar
  17. [17]
    M. E. C. Jordan, Traité des Substitutions et des Équations Algébriques, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1989.Google Scholar
  18. [18]
    A. G. Keeton, Commuting Varieties Associated with Symmetric Pairs, PhD thesis, University of California, San Diego, 1996.Google Scholar
  19. [19]
    S. H. Murray, Conjugacy classes in maximal parabolic subgroups of general linear groups, J. Algebra 233 (2000), 135–155.MathSciNetCrossRefGoogle Scholar
  20. [20]
    H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, Vol. 18, American Mathematical Society, 1999.Google Scholar
  21. [21]
    A. Premet, Nilpotent commuting varieties of reductive lie algebras, Invent. Math. 154 (2003), no. 3, 653–683.MathSciNetCrossRefGoogle Scholar
  22. [22]
    R. W. Richardson, Commuting varieties of semisimple lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327.MathSciNetzbMATHGoogle Scholar
  23. [23]
    C. M. Ringel, Iyama’s finiteness theorem via strongly quasi-hereditary algebras, J. Pure Appl. Algebra 214 (2010), no. 9, 1687–1692.MathSciNetCrossRefGoogle Scholar
  24. [24]
    G. Röhrle, On the modality of parabolic subgroups of linear algebraic groups, Manuscripta Math. 98 (1999), no. 1, 9–20.MathSciNetCrossRefGoogle Scholar
  25. [25]
    J-P. Serre, Espaces fibrés algébriques, Séminaire Claude Chevalley 3 (1958), 1–37.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ruhr-University Bochum, Faculty of Mathematics, Chair of AlgebraBochumGermany
  2. 2.Univ Lyon, Université Jean Monnet, Saint-Étienne, CNRS UMR 5208, Institut Camille JordanSaint-EtienneFrance

Personalised recommendations