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 −dimV.
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\Bibitem{Pop82}
\by V.~L.~Popov
\paper A~finiteness theorem for representations with a~free algebra of invariants
\jour Math. USSR-Izv.
\yr 1983
\vol 20
\issue 2
\pages 333--354
\mathnet{http://mi.mathnet.ru/eng/im1619}
\crossref{https://doi.org/10.1070/IM1983v020n02ABEH001353}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=651651}
\zmath{https://zbmath.org/?q=an:0547.20034}
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This publication is cited in the following 10 articles:
A. BOLSINOV, A. IZOSIMOV, I. KOZLOV, “JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS”, Transformation Groups, 28:2 (2023), 541
M. P. Bento, “The invariant space of multi-Higgs doublet models”, J. High Energ. Phys., 2021:5 (2021)
Harm Derksen, Gregor Kemper, Encyclopaedia of Mathematical Sciences, Computational Invariant Theory, 2015, 31
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D. I. Panyushev, “Regular elements in spaces of linear representations of reductive algebraic groups”, Math. USSR-Izv., 24:2 (1985), 383–390
V. L. Popov, “Syzygies in the theory of invariants”, Math. USSR-Izv., 22:3 (1984), 507–585
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