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 Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 3, Pages 544–622 (Mi izv1414)

Syzygies in the theory of invariants

V. L. Popov

Abstract: A method is developed for finding all $G$-modules (where $G$ is a connected and simply connected semisimple algebraic group over an algebraically closed field of characteristic zero) whose algebra of invariants has prescribed homological dimension. The main theorem says that the number of such $G$-modules, considered to within isomorphism and addition of a trivial direct summand, is finite. The same result is proved for finite groups $G$. All algebras of invariants of homological dimension $\leqslant10$ of a single binary form are found, as well as all algebras of invariants of a system of binary forms that are hypersurfaces. It is shown that the exceptional simple groups have no irreducible modules with an algebra of invariants of small nonzero homological dimension.
Bibliography: 46 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 22:3, 507–585

Bibliographic databases:

UDC: 519.4
MSC: Primary 15A72; Secondary 13D05

Citation: V. L. Popov, “Syzygies in the theory of invariants”, Izv. Akad. Nauk SSSR Ser. Mat., 47:3 (1983), 544–622; Math. USSR-Izv., 22:3 (1984), 507–585

Citation in format AMSBIB
\Bibitem{Pop83} \by V.~L.~Popov \paper Syzygies in the theory of invariants \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1983 \vol 47 \issue 3 \pages 544--622 \mathnet{http://mi.mathnet.ru/izv1414} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=703596} \zmath{https://zbmath.org/?q=an:0573.14003} \transl \jour Math. USSR-Izv. \yr 1984 \vol 22 \issue 3 \pages 507--585 \crossref{https://doi.org/10.1070/IM1984v022n03ABEH001455} 

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This publication is cited in the following articles:
1. Akihiko Gyoja, “Invariants, nilpotent orbits, and prehomogeneous vector spaces”, Journal of Algebra, 142:1 (1991), 210
2. Shmel'kin D.A., “On algebras of invariants and codimension 1 Luna strata for nonconnected groups”, Geometriae Dedicata, 72:2 (1998), 189–215
3. Philippe Pouliot, J Phys A Math Gen, 34:41 (2001), 8631
4. Shmel'kin D.A., “On representations of SLn with algebras of invariants being complete intersections”, Journal of Lie Theory, 11:1 (2001), 207–229
5. Bibikov P.V., Lychagin V.V., “Klassifikatsiya lineinykh deistvii algebraicheskikh grupp na prostranstvakh odnorodnykh form”, Doklady akademii nauk, 442:6 (2012), 732–732
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