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Trudy Mat. Inst. Steklova, 2003, Volume 241, Pages 192–209 (Mi tm396)  

This article is cited in 16 scientific papers (total in 16 papers)

The Cone of Hilbert Nullforms

V. L. Popov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A geometric–combinatorial algorithm is given that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the nullcone of a linear representation of a reductive algebraic group and compute their dimensions. In particular, it provides a constructive approach to computing the dimension of the nullcone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, $\theta$-representation), the algorithm turns into the algorithm of classifying conjugacy classes of nilpotent elements in a semisimple Lie algebra (respectively, homogeneous nilpotent elements in a cyclically graded semisimple Lie algebra).

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 241, 177–194

Bibliographic databases:
UDC: 512.745
Received in December 2002

Citation: V. L. Popov, “The Cone of Hilbert Nullforms”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 241, Nauka, MAIK Nauka/Inteperiodika, M., 2003, 192–209; Proc. Steklov Inst. Math., 241 (2003), 177–194

Citation in format AMSBIB
\Bibitem{Pop03}
\by V.~L.~Popov
\paper The Cone of Hilbert Nullforms
\inbook Number theory, algebra, and algebraic geometry
\bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich
\serial Trudy Mat. Inst. Steklova
\yr 2003
\vol 241
\pages 192--209
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm396}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2024052}
\zmath{https://zbmath.org/?q=an:1125.14301}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 241
\pages 177--194


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Popov V.L., Tevelev E.A., “Self-dual projective algebraic varieties associated with symmetric spaces”, Algebraic Transformation Groups and Algebraic Varieties, Encyclopedia of Mathematical Sciences, 2004, 131–167  crossref  mathscinet  zmath  isi
    2. Draisma J., “Counting components of the null–cone on tuples”, Transformation Groups, 11:4 (2006), 609–624  crossref  mathscinet  zmath  isi  scopus
    3. Kraft H., Wallach N.R., “On the nullcone of representations of reductive groups”, Pacific Journal of Mathematics, 224:1 (2006), 119–139  crossref  mathscinet  zmath  isi  scopus
    4. Kostant B., “On the centralizer of K in U (g)”, Journal of Algebra, 313:1 (2007), 252–267  crossref  mathscinet  zmath  isi  scopus
    5. de Graaf W.A., “Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type”, Lms Journal of Computation and Mathematics, 11 (2008), 280–297  crossref  mathscinet  zmath  isi
    6. V. L. Popov, “Two Orbits: When Is One in the Closure of the Other?”, Proc. Steklov Inst. Math., 264 (2009), 146–158  mathnet  crossref  mathscinet  isi  elib  elib
    7. Kato S., “An Exotic Deligne–Langlands Correspondence for Symplectic Groups”, Duke Mathematical Journal, 148:2 (2009), 305–371  crossref  mathscinet  zmath  isi  scopus
    8. Charbonnel J.-Y., Moreau A., “Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra”, Transformation Groups, 14:2 (2009), 319–360  crossref  mathscinet  zmath  isi  elib  scopus
    9. de Graaf W.A., “Computing representatives of nilpotent orbits of theta-groups”, J Symbolic Comput, 46:4 (2011), 438–458  crossref  mathscinet  zmath  isi  scopus
    10. Clarke M.C., “Computing nilpotent and unipotent canonical forms: a symmetric approach”, Math Proc Cambridge Philos Soc, 152:1 (2012), 35–53  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. Zaiter M., “On Related Varieties to the Commuting Variety of a Semisimple Lie Algebra”, J. Algebra, 376 (2013), 10–24  crossref  mathscinet  zmath  isi  elib  scopus
    12. V. L. Popov, “Number of components of the nullcone”, Proc. Steklov Inst. Math., 290:1 (2015), 84–90  mathnet  crossref  crossref  isi  elib  elib
    13. de Graaf W.A., “Orbit Closures of Linear Algebraic Groups”, Computer Algebra and Polynomials, Lecture Notes in Computer Science, 8942, ed. Gutierrez J. Schicho J. Weimann M., Springer-Verlag Berlin, 2015, 76–93  crossref  mathscinet  zmath  isi  scopus
    14. Derksen H. Kemper G., “Is One of the Two Orbits in the Closure of the Other?”: Derksen, H Kemper, G, Computational Invariant Theory, 2Nd Edition, Encyclopaedia of Mathematical Sciences, 130, Springer-Verlag Berlin, 2015, 309–322  mathscinet  isi
    15. Derksen H., Kemper G., “Stratification of the Nullcone”: Derksen, H Kemper, G, Computational Invariant Theory, 2Nd Edition, Encyclopaedia of Mathematical Sciences, 130, Springer-Verlag Berlin, 2015, 323–343  mathscinet  isi
    16. Nishiyama K., Ohta T., “Enhanced Adjoint Actions and Their Orbits For the General Linear Group”, Pac. J. Math., 298:1 (2019), 141–155  crossref  isi
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