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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 2, Pages 347–370 (Mi izv1619)  

This article is cited in 4 scientific papers (total in 4 papers)

A finiteness theorem for representations with a free algebra of invariants

V. L. Popov


Abstract: It is proved that for any connected semisimple algebraic group $G$ defined over an algebraically closed field of characteristic zero there exist (up to isomorphism) only a finite number of finite-dimensional rational $G$-modules containing no nonzero fixed vectors and having a free algebra of invariants. The proof is constructive and makes it possible in principle to indicate these $G$-modules explicitly. It is also proved that for all irreducible $G$-modules $V$, except for a finite number, the degree of the Poincaré series of the algebra of invariants (regarded as a rational function) equals $-\dim V$.
Bibliography: 21 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 20:2, 333–354

Bibliographic databases:

UDC: 519.4
MSC: Primary 15A72, 20G05; Secondary 52A25
Received: 14.09.1981

Citation: V. L. Popov, “A finiteness theorem for representations with a free algebra of invariants”, Izv. Akad. Nauk SSSR Ser. Mat., 46:2 (1982), 347–370; Math. USSR-Izv., 20:2 (1983), 333–354

Citation in format AMSBIB
\Bibitem{Pop82}
\by V.~L.~Popov
\paper A~finiteness theorem for representations with a~free algebra of invariants
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1982
\vol 46
\issue 2
\pages 347--370
\mathnet{http://mi.mathnet.ru/izv1619}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=651651}
\zmath{https://zbmath.org/?q=an:0547.20034}
\transl
\jour Math. USSR-Izv.
\yr 1983
\vol 20
\issue 2
\pages 333--354
\crossref{https://doi.org/10.1070/IM1983v020n02ABEH001353}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. L. Popov, “Syzygies in the theory of invariants”, Math. USSR-Izv., 22:3 (1984), 507–585  mathnet  crossref  mathscinet  zmath
    2. D. I. Panyushev, “Regular elements in spaces of linear representations of reductive algebraic groups”, Math. USSR-Izv., 24:2 (1985), 383–390  mathnet  crossref  mathscinet  zmath
    3. N Alon, K.A Berman, “Regular hypergraphs, Gordon's lemma, Steinitz' lemma and invariant theory”, Journal of Combinatorial Theory, Series A, 43:1 (1986), 91  crossref
    4. D. I. Panyushev, “Orbits of maximal dimension of solvable subgroups of reductive linear groups, and reduction for $U$-invariants”, Math. USSR-Sb., 60:2 (1988), 365–375  mathnet  crossref  mathscinet  zmath
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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