Invariant Theory

  • V. L. Popov
  • E. B. Vinberg
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 55)

Abstract

The problems being solved by invariant theory are far-reaching generalizations and extensions of problems on the “reduction to canonical form” of various objects of linear algebra or, what is almost the same thing, projective geometry.

Keywords

Manifold Assure Stratification Hull Smoke 

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References

  1. Monographs on classical invariant theory: Elliot (1913), Fogarty (1969), Grace and Young (1903), Gurevich (1948b), Procesi (1982), Schur (1968), Spencer (1971), Springer (1977), Weitzenböck (1923).Google Scholar
  2. Monographs on modern invariant theory: Dieudonné and Carrell (1971), Fogarty (1969), Kraft (1984), Kraft, Petrie, and Schwarz (1989), Kraft, Slodowy, and Springer (1989), Mumford (1965), Mumford and Fogarty ( 1982 ), V.L. Popov (1992a).Google Scholar
  3. Surveys: Kung and Rota (1984), Parshin (1974), V.L. Popov (1979a), (19866), (1991), Vinberg (1982b).Google Scholar
  4. Invariant theory of binary forms: Alekseev (1899), Dickson (1914), (1966), Dixmier and Lazard (1985/86), Elliot (1913), Gordan (1987), Grace and Young (1903), Gurevich (1948b), Khadzhiev (1978), Shioda (1967), Springer (1977).Google Scholar
  5. Invariants and orbits of specific linear groups: Beklemishev (1981), (1982), Gurevich (1948b), Kraft (1984) (cubic forms in three or four variables); Bui Viet Kha (1982), Kuribayashi and Koimya (1979), Katsylo (1992), (1993) (forms of degree 4 in three variables); Hodge and Pedoe (1947), Spaltenstein (1983) (bilinear forms); Adamovich and Golovina (1977), Gantmakher (1966), Gurevich (1950), Spaltenstein (1983) (pairs of bilinear forms); Egorov (1981), Gurevich (1948a), Katanova (1992), Vinberg and Ehlashvili (1978) (trivectors); Antonyan (1981), Katanova (1992) (four-vectors); Antonyan and Ehlashvili (1982), Gatti and Viniberghi (1978), Igusa (1970), V.L. Popov (1978b) (spinors); Kac (1983), Schoefield (1991), (1992), Adamovich and Golovina (1979), Djocovié (1988), Igusa (1973), Kimura and Kasai (1985), Newstead (1982), Panyushev (1982a), Schwarz (1978a), (1983), (1987), (1988), Sibirskij (1976), (1982), Simoniya (1960), Thrall and Chanler (1938).Google Scholar
  6. Invariants and orbits of adjoint representations and isotropy representations of symmetric spaces: Alekseevskij (1979), Antonyan (1982), Borho (1981), Borho and Kraft (1979), Djocovié (1988), Doan Quynh (1968), Donkin (1988), Dynkin (1952), Ehlashvili (1975), (1984), Ehlashvili, Grélaud (1993), Katsylo (1982), Kempken (1982), Kostant (1963), Kostant and Rallis (1971), Kraft and Procesi (1979), (1982), Panyushev (1991), (1993d), Peterson (1978), Spaltenstein (1982), Springer (1976), Weyman (1989).Google Scholar
  7. Linear groups whose algebra of invariants is a complete intersection: Gordeev (1986), Nakajima (1984), (1985), (1986), Nakajima and Watanabe (1984), Stanley (1979).Google Scholar
  8. Invariants and orbits of Borel and maximal unipotent subgroups: Brion (1985), Ehlashvili (1987), Khadzhiev (1967), Littelmann (1993), Panyushev (1987), (1990), (1992), (1993b), (1993c), V.L. Popov (1986a), (1988), Vinberg (1986), Vinberg and Kimel’fel’d (1978).Google Scholar
  9. Invariants of other non-reductive groups and the fourteenth Hilbert problem: A’Campo (1992), Aleksandrov (1969), Browder (1976), Choodnovsky (1978), Donkin (1988b), Grosshans (1973), (1983), (1986), Hochschild and Moston (1973), Horvath (1989a), (1989b), Humphreys (1978), Knop (1992), Miyata (1971), Nagata (1959), Pommerening (1981), (1987a), (1987b), V.L. Popov (19796), (1992a), Roberts (1990), Tan (1988), (1989a), (1989b), (1992), Veles’ko (1985), (1986), Vinberg (1982), Weitzenböck (1932), Zalesskij (1983).Google Scholar
  10. Locally transitive actions, in particular, linear representations, and spherical homogeneous spaces: Akhiezer (1977), (1979), (1985), (1986), Brion (1987a), Brion and Luna (1987), Brion and Pauer (1987), De Concini and Procesi (1983), (1985), Igusa (1970), Kac (1980), Kempf et al. (1973), Kimura and Kasai (1985), Kimura, Kasai, and Yasukura (1986), Knop (1991), (1993), Kraft (1984), Littelman (1993), Luna (1993), Luna and Vust (1983), Mikityuk (1986), Muller, Rubenthaler, and Schiffmann (1986), Panyushev (1993d), V.L. Popov (1973), Pyasetskij (1975), (1983), (1985), Sato and Kimura (1974), (1977), Servedio (1973), Shpiz (1978), Thrall and Chanler (1938), Vinberg (1960), (1986), Vinberg and Kimel’fel’d (1978), Vinberg and Popov (1972), Vust (1987).Google Scholar
  11. Invariant theory over fields of positive characteristic: Bardsley and Richardson (1985), De Concini and Procesi (1976), Dickson (1966), Fogarty (1983), Haboush (1975), Mumford (1965), Nagata (1964), (1965), Premet (1991), Procesi (1976), Richardson (1976), (1977), Richmond (1990), Seshadri (1972), (1977), Slodowy (1980), Springer (1977), Steinberg (1974).Google Scholar
  12. Invariant theory over algebraically nonclosed fields: Birkes (1971), De Concini and Procesi (1976), Igusa (1970), Procesi (1982), Procesi and Schwarz (1985), Rousseau (1978), Voskresenskij (1977)Google Scholar
  13. Adamovich, O.M. [1980]: Equidimensional representations of simple algebraic groups. Geom. Metod. Zadach. Algebry Anal. 2, 120–125.Google Scholar
  14. Adamovich, O.M. [1980]: English transi.: Transi., II. Ser., Ann. Math. Soc. 128, 25–29 [1986]. Zb1.463.14017Google Scholar
  15. Adamovich, O.M., Golovina, E.O. [1977]: On the invariants of a pair of bilinear forms. Vestn. Mosk. Univ., Ser. I 1977, No. 2, 15–18.MATHGoogle Scholar
  16. Adamovich, O.M., Golovina, E.O. [1977]: English transi.: Mosc. Univ. Math. Bull. 32, No. 2, 11–14 [1977]. Zb1.365.15011MATHGoogle Scholar
  17. Adamovich, O.M., Golovina, E.O. [1979]: Simple linear Lie groups having a free algebra of invariants. In: Vopr. Teor. Grupp. Gomologicheskoj Algebry 2, 3–41.Google Scholar
  18. Adamovich, O.M., Golovina, E.O. [1979]: English transi.: Sel. Math. Sov. 3, No. 2, 183–220 [1984]. Zb1.446.22017MATHGoogle Scholar
  19. Adams, J.F. [1969]: Lectures on Lie Groups. New York, Amsterdam: W.A. Benjamin. 130 pp. Zb1.206,316Google Scholar
  20. Akhiezer, D.N. [1977]: Dense orbits with two endpoints. Izv. Akad. Nauk SSSR, Ser. Mat. 41, No. 2, 308–324.MATHGoogle Scholar
  21. Akhiezer, D.N. [1977]: English transi.: Math. USSR. Izv. 11, 293–307 [1977]. Zb1.373.14016Google Scholar
  22. Akhiezer, D.N. [1979]: Algebraic groups acting transitively in the complement of a homogeneous hypersurface. Dokl. Akad. Nauk SSSR 245, No. 2, 281–284.Google Scholar
  23. Akhiezer, D.N. [1979]: English transi.: Sov. Math., Dokl. 20, 278–291 [1979]. Zb1.437.14027Google Scholar
  24. Akhiezer, D.N. [1985]: On actions with a finite number of orbits. Funkts. Anal. Prilozh. 19, No. 1, 1–5.MATHGoogle Scholar
  25. Akhiezer, D.N. [1985]: English transi: Funct. Anal. Appl. 19, 1–4 [1985]. Zb1.576.14045Google Scholar
  26. Akhiezer, D.N. [1986]: Homogeneous Complex Manifolds. Itogi Nauki Tekh., Ser. Sovrem. Probi. Mat., Fundam. Napravleniya 10, 223–275.Google Scholar
  27. Akhiezer, D.N. [1986]: English transi. in: Several Complex Variables IV, Encycl. Math. Sci. 10, 195–244.Google Scholar
  28. Aleksandrov, P.S. [ed.] [1969]: Hilbert’s Problems. Moscow: Nauka, 240 pp. [Russian]Google Scholar
  29. Alekseev, V.G. [1899]: Theory of Rational Invariants of Binary Forms. Dorpat Univ., 232 pp. [Russian]. FdM 30, 110Google Scholar
  30. Alekseevskij, A.V. [1979]: Component groups of centralizers of unipotent elements in semisimple algebraic groups. Tr. Tbilis. Mat. Inst. Razmadze 62, 5–27. Zb1.455.20033MATHGoogle Scholar
  31. Andreev, E.M., Popov, V.L. [1971]: Stationary subgroups of points in general position in the representation space of a semisimple Lie group. Funkts. Anal. Prilozh. 5, No. 4, 1–8. English transi.: Funct. Anal. Appl. 5, 265–271 [1972]. Zb1.246.22017MATHGoogle Scholar
  32. Andreev, E.M., Vinberg, E.B., Ehlashvili, A.G. [1967]: Orbits of highest dimension of semisimple linear Lie groups. Funkts. Anal. Prilozh. 1, No. 1, 3–7.MATHGoogle Scholar
  33. Andreev, E.M., Vinberg, E.B., Ehlashvili, A.G. [1967]: English transi.: Funct. Anal. Appl. 1, 257–261 [1968]. Zb1.176,303Google Scholar
  34. Antonyan, L.V. [1981]: Classification of four-vectors of an eight-dimensional space. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 20, 144–161. Zb1.467.15018Google Scholar
  35. Antonyan, L.V. [1982]: On the classification of homogeneous elements of Z2 -graded semisimple Lie algebras. Vestn. Mosk. Univ., Ser. I, No. 2, 29–34.Google Scholar
  36. Antonyan, L.V. [1982]: English transi.: Mosc. Univ. Math. Bull. 37, No. 2, 36–43 [1982]. Zb1.494.17008MATHGoogle Scholar
  37. Antonyan, L.V. [1987]: On homogeneous nilpotent elements of periodically graded semisimple Lie algebras. In: Vopr. Teor. Grupp Gomologicheskoj Algebry 7, 55–64. Zb1.706.17015Google Scholar
  38. Antonyan, L.V., Ehlashvili, A.G. [1982]: Classification of spinors of dimension sixteen. Tr. Tbilis. Mat. Inst. Razmadze 70, 5–23. Zb1.519.17006MATHGoogle Scholar
  39. Armstrong, M.A. [1968]: On the fundamental group of an orbit space of a discontinuous group. Proc. Camb. Philos. Soc. 64, 299–301. Zb1.159,530MATHGoogle Scholar
  40. Arnol’d, V.I. [1972]: Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek, and Lagrangian singularities. Funkts. Anal. Prilozh. 6, No. 4, 3–25. English transi.: Funct. Anal. Appl. 6, 254–272 [1973]. Zb1.278.57011Google Scholar
  41. Arnol’d, V.I., Givental’, A.B. [1985]: Symplectic Geometry. Itogi Nauki Tekh., Ser. Sovrem. Probi. Mat., Fundam. Napravleniya 4, 7–39.Google Scholar
  42. Arnol’d, V.I., Givental’, A.B. [1985]: English transi. in: Dynamical Systems IV: Encycl. Math. Sci. 4, 1–136. Berlin, Heidelberg, New York: Springer-Verlag 1990. Zb1.592.58030Google Scholar
  43. Bardsley, P., Richardson, R.W. [1985]: Etale slices for algebraic transformation groups in characteristic p. Proc. Lond. Math. Soc., III. Ser. 51, No. 3, 295–317. Zb1.604.14037MATHGoogle Scholar
  44. Beklemishev, N.D. [1981]: Classification of quaternary cubic forms not in general position. In: Vopr. Teor. Grupp Gomologicheskoj Algebry 1981, 3–17.Google Scholar
  45. Beklemishev, N.D. [1981]: English transi.: Sel. Math. Sov. 5, 203–218 [1986]. Zb1.482.14014Google Scholar
  46. Beklemishev, N.D. [1982]: Invariants of cubic forms in four variables. Vestn. Mosk. Univ., Ser. I, No. 2, 42–49.Google Scholar
  47. Beklemishev, N.D. [1982]: English transi.: Mosc. Univ. Math. Bull. 37, No. 2, 54–62 [1982]. Zb1.498.14006MATHGoogle Scholar
  48. Bessenrodt, Ch., Le Bruyn, L. [1991]: Stable rationality of certain PGL„ quotients. Invent. Math. 104, 179–199. Zb1.741.14032MATHGoogle Scholar
  49. Bialynicki-Birula, A. [1966]: Remarks on the action of an algebraic torus on V. I. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 14, 177–181. Zb1.163,429Google Scholar
  50. Bialynicki-Birula, A. [1967]: Remarks on the action of an algebraic torus on k“. II. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 15, 123–125. Zb1.163,429Google Scholar
  51. Bialynicki-Birula, A. [1973]: Some theorems on actions of algebraic groups. Ann. Math., II. Ser. 98, 480–497. Zb1.275.14007MATHGoogle Scholar
  52. Bialynicki-Birula, A. [1976]: Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24, 667–674. Zb1.355.14015MATHGoogle Scholar
  53. Bialynicki-Birula, A., Hochschild, G., Mostow, G.D. [1963]: Extensions of representations of algebraic linear groups. Am. J. Math. 85, 131–144. Zb1.116,23MATHGoogle Scholar
  54. Birkes, D. [1971]: Orbits of linear algebraic groups. Ann. Math., II. Ser. 93, No. 3, 459–475. Zb1.198,350Google Scholar
  55. Bogomolov, F.A. [1978]: Holomorphic tensors and vector bundles on projective varieties. Izv. Akad. Nauk SSSR, Ser. Mat. 42, No. 6, 1227–1287.Google Scholar
  56. Bogomolov, F.A. [1978]: English transi.: Math. USSR, Izv. 13, 499–555 [1979]. Zb1.439.14002Google Scholar
  57. Bogomolov, F.A., Katsylo, P.I. [1985]: Rationality of certain quotient varieties. Mat. Sb. Nov. Ser. 126, No. 4, 584–591.Google Scholar
  58. Bogomolov, F.A., Katsylo, P.I. [1985]: English transi.: Math. USSR, Sb. 54, 571–576 [1986]. Zb1.591.14040Google Scholar
  59. Borel, A., Tits, J. [1971]: Eléments unipotents et sous-groupes paraboliques de groupes réductifs I. Invent. Math. 12, 95–104MATHGoogle Scholar
  60. Borho, W. [1981]: Über Schichten halbeinfacher Lie-Algebren. Invent. Math. 65, 283–317. Zb1.484.17004MATHGoogle Scholar
  61. Borho, W. Kraft, H. [1979]: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54, 61–104. Zb1.395.14013MATHGoogle Scholar
  62. Bourbaki, N. [1959]: Algèbre. Chapitre 9. Paris: Hermann. 258 pp. Zb1.102,255Google Scholar
  63. Bourbaki, N. [1968]: Groupes et Algèbres de Lie. Chapitres 4–6. Paris: Hermann. 288 pp. Zb1.186,330Google Scholar
  64. Boutot, J-F. [1987]: Singularités rationnelles et quotients par les groupes réductifs. Invent. Math. 88, 65–68. Zb1.619.14029MATHGoogle Scholar
  65. Bredon, G.E. [1972]: Introduction to Compact Transformation Groups. New York, London: Academic Press. 459 pp. Zb1.246.57017Google Scholar
  66. Brion, M. [1985]: Représentations exceptionnelles des groupes semisimples. Ann. Sci. Ec. Norm. Super., IV. Ser. 18, 345–387. Zb1.588.22010MATHGoogle Scholar
  67. Brion, M. [1987a]: Classification des espaces homogènes sphériques. Compos. Math. 63, 189–208. Zb1.642.14011MATHGoogle Scholar
  68. Brion, M. [1987b]: Sur l’image de l’application moment. Lect. Notes Math. 1296, 177–192. Zb1.667.58012Google Scholar
  69. Brion, M., Luna, D., Vust, T. [1986]: Espaces homogènes sphériques. Invent. Math. 84, 617–632. Zb1.604.14047MATHGoogle Scholar
  70. Brion, M., Luna D. [1987]: Sur la structure locale des variétés sphériques. Bull. Soc. Math. France 115, 211–226MATHGoogle Scholar
  71. Brion, M., Pauer, F. [1987]: Valuations des espaces homogènes sphériques. Comm. Math. Heiv. 62, 265–285. Zb1.627.14038MATHGoogle Scholar
  72. Broer, B. [1990]: Hilbert series in invariant theory. Ph.D. Thesis. Rijksuniversiteit te Utrecht, 103 pp.Google Scholar
  73. Broer, B. [1993a]: Classification of Cohen—Macaulay modules of covariants for systems of binary forms [to appear in Proc. AMS]Google Scholar
  74. Broer, B. [1993b]: A new method for calculating Hilbert series [to appear in J. Algebra]Google Scholar
  75. Broer, B. [1993c]: Hilbert series for modules of covariants [to appear in Proc. Symp. Pure Math.]Google Scholar
  76. Brouwer, A.E., Cohen, A.M. [1979]: The Poincaré series of the polynomials invariant under SU 2 in its irreducible representation of degree 17. Math. Cent. Amsterdam, Afd. Zuivere Viskd. ZW. 134/79, 1–20. Zb1.417.22008Google Scholar
  77. Browder, F.E. [ed.] [1976]: Mathematical Developments Arising From Hilbert Problems. Proc. Symp. Pure Math. 28, Providence, R.I.: American Mathematical Society. 628 pp. Zb1.326.00002Google Scholar
  78. Bui Viet Kha [1982]: Classification of ternary forms of fourth degree with nontrivial group of automorphisms. In: Vopr. Teor. Grupp. Gomologicheskoj Algebry 1982, 150–153. [Russian]. Zb1.567.20028Google Scholar
  79. Chevalley, C. [1951]: Théorie des Groupes de Lie. Tome 2. Paris: Hermann. 196 pp. Zb1.54,13Google Scholar
  80. Chevalley, C. [1955]: Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782. Zb1.65,261MATHGoogle Scholar
  81. Chevalley, C. [1958]: Fondements de la Géométrie Algébrique. Paris: Secrétariat Mathématique. 222 pp. Zb1.87,855Google Scholar
  82. Choodnovsky, G. [1978]: Sur la construction de Rees et Nagata pour le 14-e problème de Hilbert. C.R. Acad. Sci. Paris 286, A1133–1135Google Scholar
  83. Colliot-Théléne, J.-L., Ojanguren, M. [1992]: Espaces homogènes localement triviaux. Publ. Math. IHES 75, 97–121MATHGoogle Scholar
  84. Dadok, J., Kac, V. [1985]: Polar representations. J. Algebra 92, No. 2, 504–524. Zbl.611.22009MATHGoogle Scholar
  85. De Concini, C., Procesi, C. [1976]: A characteristic free approach to invariant theory. Adv. Math. 21, 330–354. Zb1.347.20025MATHGoogle Scholar
  86. De Concini, C., Procesi, C. [1983]: Complete symmetric varieties. Lect. Notes Math. 996, 1–44. Zb1.581.14041Google Scholar
  87. De Concini, C., Procesi, C. [1985]: Complete symmetric varieties. II. Adv. Stud. Pure Math. 6, 481–513. Zb1.596.14041Google Scholar
  88. Dickson, L.E. [1914]: Algebraic Invariants. New York: Wiley. 100 pp. FdM 45, 196Google Scholar
  89. Dickson, L.E. [1966]: On Invariants and the Theory of Numbers. New York: Dover. 110 pp. Zb1.139,266Google Scholar
  90. Dieudonné, J.A., Carrell, J.B. [1971]: Invariant Theory. New York, London: Academic Press. Zb1.258.14011Google Scholar
  91. Dixmier, J. [1975]: Polarisations dans les algèbres de Lie semisimples complexes. Bull. Sci. Math., II. Ser. 99, 45–63. Zb1.314.17009MATHGoogle Scholar
  92. Dixmier, J. [1987]: Quelques résultats de finitude en théorie des invariants [d’après V.L. Popov]. Sémin. Bourbaki: 1985/86, Astérisque 145/146, 163–175. Zb1.618.20026Google Scholar
  93. Dixmier, J., Erdös, P., Nicolas, J-L. [1987]: Sur le nombre d’invariants fondamentaux des formes binaires. C.R. Acad. Sci., Paris, Sér. I 305, 319–322. Zb1.642.10021Google Scholar
  94. Dixmier, J., Lazard, D. [1985/86]: Le nombre minimum d’invariants fondamentaux pour les formes binaires de degré 7. Port. Math. 43, No. 3, 377–392. Zb1.602.15022Google Scholar
  95. Djokovic, D.Z. [1988]: Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers. J. Algebra 112, No. 2, 503–524. Zb1.639.17005MATHGoogle Scholar
  96. Doan Quynh [1968]: Poincaré polynomials of compact homogeneous Riemann spaces with an irreducible stationary subgroup. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 14, 33–93. Zb1.187,438Google Scholar
  97. Dolgachev, I.V. [1987]: Rationality of fields of invariants. Proc. Symp. Pure Math. 46, 3–16. Zb1.659.14009Google Scholar
  98. Donaldson, S. [1984]: Instantons and geometric invariant theory. Commun. Math. Phys. 93, 453–460. Zb1.581.14008MATHGoogle Scholar
  99. Donkin, S. [1988a]: On conjugating representations and adjoint representations of semisimple groups. Invent. Math. 91, 137–145. Zb1.639.20021MATHGoogle Scholar
  100. Donkin, S. [1988b]: Invariants of unipotent radicals. Math. Z. 198, 117–125MATHGoogle Scholar
  101. Dynkin, E.B. [1952]: Semisimple subalgebras of semisimple Lie algebras. Mat. Sb., Nov. Ser. 30, No. 2, 349–462.Google Scholar
  102. Dynkin, E.B. [1952]: English transi.: Transi., II. Ser., Am. Math. Soc. 6, 111–243 [1957]. Zb1.48,17Google Scholar
  103. Egorov, G.V. [1981]: Invariants of a trivector in a nine-dimensional space. In: Vopr. Teor. Grupp Gomologicheskoj Algebry 1981, 127–131. Zb1.481.15014Google Scholar
  104. Ehlashvili, A.G. [1972a]: A canonical form and stationary subalgebras of points in general position for simple linear Lie groups. Funkts. Anal. Prilozh. 6, No. 1, 51–62. English transi.: Funct. Anal. Appl. 6, 44–53 [1972]. Zb1.252.22015Google Scholar
  105. Ehlashvili, A.G. [1972b]: Stationary subalgebras of points in general position for irreducible linear Lie groups. Funkts. Anal. Prilozh. 6, No. 2, 65–78. English transl.: Funct. Anal. Appl. 6, 139–148 [1972]. Zb1.252.22016Google Scholar
  106. Ehlashvili, A.G. [1975]: Centralizers of nilpotent elements in semisimple Lie algebras. Tr. Tbilis. Mat. Inst. Razmadze 46, 109–132 [Russian]. Zb1.323.17004Google Scholar
  107. Ehlashvili, A.G. [1984]: Sheets of simple Lie algebras of exceptional type. In: Studies in Algebra, Tbilisi, 171–194 [Russian]Google Scholar
  108. Ehlashvili, A.G. [1987]: Orbits of maximal dimension for Borel subgroups of semisimple linear Lie groups. Funkts. Anal. Prilozh. 21, No. 1, 92–93. English transi.: Funct. Anal. Appl. 21, 84–86 [1987]. Zb1.618.20032Google Scholar
  109. Ehlashvili, A.G. [1992]: Invariant algebras. Adv. Sov. Math. 8 [Vinberg, E.B., ed.] Amer. Math. Soc., 57–64Google Scholar
  110. Ehlashvili, A.G., Grélaud, G. [1993]: Classification des éléments nilpotents compacts des algèbres de Lie simples. C.R. Acad. Sci. Paris [to appear]Google Scholar
  111. Elliot, E.B. [1913]: An Introduction to the Algebra of Quantics. London: Clarendon Press. FdM 44, 155Google Scholar
  112. Fogarty, J. [1969]: Invariant Theory. New York, Amsterdam: W.A. Benjamin. 216 pp. Zb1.191,517Google Scholar
  113. Fogarty, J. [1983]: Geometric quotients are algebraic schemes. Adv. Math. 48, 166–171. Zb1.556.14023MATHGoogle Scholar
  114. Formanek, E. [1979]: The center of the ring of 3 by 3 generic matrices. Linear Algebra 7, 203–212MATHGoogle Scholar
  115. Formanek, E. [1980]: The center of the ring of 4 by 4 generic matrices. J. Algebra 62, 304–319MATHGoogle Scholar
  116. Fossum, R.M. [1981]: Invariant theory, representation theory, commutative algebra—ménage à trois, Lect, Notes Math. 867, 1–37. Zb1.466.14004Google Scholar
  117. Gantmakher, F.R. [1966]: Theory of Matrices [2nd ed.]. Moscow: Nauka. 576 pp. French, transi.: Paris: Dunod 1966. Zb1.145,36; Zb1.50,248Google Scholar
  118. Gatti, V. [Kac, V.G.], Viniberghi, E. [Vinberg, E.B.] [1978]: Spinors of 13-dimensional space. Adv. Math. 30, 137–155. Zb1.429.20043MATHGoogle Scholar
  119. Gieseker, D. [1977]: Global moduli for surfaces of general type. Invent. Math. 43, 233–282. Zb1.389.14006MATHGoogle Scholar
  120. Gieseker, D. [1983]: Geometric invariant theory and applications to moduli problems. Lect. Notes Math. 996, 45–73. Zb1.582.14001Google Scholar
  121. Gordan, P. [1987]: Invariantentheorie, Chelsea Publishing Company, New York.MATHGoogle Scholar
  122. Gordeev, N.L. [1986]: Finite linear groups whose algebra of invariants is a complete intersection. Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 2, 343–392.MATHGoogle Scholar
  123. Gordeev, N.L. [1986]: English transi.: Math. USSR, Izv. 28, 335–379 [1987]. Zb1.606.14007Google Scholar
  124. Grace, J.H., Young, A. [1903]: The Algebra of Invariants. Cambridge: University Press. 384 pp. Reprinted: Bronx, New York: Chelsea. FdM. 34,114Google Scholar
  125. Grosshans, F. [1973]: Observable groups and Hilbert’s fourteenth problem. Am. J. Math. 95, No. 1, 229–253. Zb1.309.14039MATHGoogle Scholar
  126. Grosshans, F. [1983]: The invariants of unipotent radicals of parabolic subgroups. Invent. Math. 73, 1–9MATHGoogle Scholar
  127. Grosshans, F. [1986]: Hilbert’s fourteenth problem for non-reductive groups. Math. Z. 193, 95–103MATHGoogle Scholar
  128. Grothendieck, A. [1959/60]: Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. Sémin. Bourbaki, Exp. 190. Zb1.229.14007Google Scholar
  129. Grothendieck, A. [1960/61]: ibid. IV. Les schémas de Hilbert. Sémin. Bourbaki, Exp. 221. Zb1.236.14003Google Scholar
  130. Guillemin, V., Sternberg, S. [1982]: Convexity properties of the moment mapping. I. Invent. Math. 67, No. 3, 491–513. Zb1.503.58017MATHGoogle Scholar
  131. Guillemin, V., Sternberg, S. [1984]: Convexity properties of the moment mapping. II. Invent. Math. 77, No. 3, 533–546. Zb1.561.58015MATHGoogle Scholar
  132. Gurevich, G.B. [1948a]: The algebra of a trivector. II. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 6, 28–124.Google Scholar
  133. Gurevich, G.B. [1948b]: Fundamentals of the Theory of Algebraic Invariants. Moscow: Gostekhizdat. 408 pp. [Russian]Google Scholar
  134. Gurevich, G.B. [1950]: Canonization of a pair of bivectors. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 8, 355–363 [Russian]. Zb1.41,484Google Scholar
  135. Haboush, W. [1975]: Reductive groups are geometrically reductive. Ann. Math., II. Ser. 102, 67–83. Zb1.316.14016MATHGoogle Scholar
  136. Hajja, M. [1983]: Rational invariants of meta-abelian groups of linear automorphisms. J. Algebra 80, No. 2, 295–305. Zb1.544.20007MATHGoogle Scholar
  137. Harish-Chandra [1958]: Spherical functions on a semi-simple Lie group. I, II. Am. J. Math. 80, 241–310, 553–613. Zb1.93,128Google Scholar
  138. Hartshorne, R. [1977]: Algebraic Geometry. New York: Springer. 496 pp. Zb1.367.14001Google Scholar
  139. Helgason, S. [1978]: Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press. 628 pp. Zb1.451.53038Google Scholar
  140. Hesselink, W.H. [1978]: Uniform instability in reductive groups. J. Reine Angew. Math. 303/304, 74–96. Zb1.386.20020Google Scholar
  141. Hilbert, D. [1890]: Über die Theorie der algebraischen Formen. Math. Ann. 36, 473–534 FdM22,133MATHGoogle Scholar
  142. Hilbert, D. [1893]: Über die vollen Invariantensysteme. Math. Ann. 42, 313–373. FdM25,173MATHGoogle Scholar
  143. Hochschild, G., Mostow, G.D. [1973]: Unipotent groups in invariant theory. Proc. Nat. Acad. Sci. U.S.A. 70, 646–648Google Scholar
  144. Hochster, M. [1985]: Invariant theory of commutative rings. Contemp. Math. 43, 161–179. Zb1.592.14009Google Scholar
  145. Hochster, M., Huneke, C. [1988]: Tightly closed ideals. Bull. Am. Math. Soc., New Ser. 18, No. 1, 45–48. Zb1.674.13003MATHGoogle Scholar
  146. Hochster, M., Roberts, J. [1974]: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. Math. 13, 125–175. Zb1.289.14010Google Scholar
  147. Hodge, W.V.D., Pedoe, D. [1947]: Methods of Algebraic Geometry. Vol. 1. Cambridge: University Press. 440 pp.Google Scholar
  148. Horvath, J. [1989a]: Weight spaces of invariants of certain unipotent group actions. J. Algebra 126, 293–299Google Scholar
  149. Horvath, J. [1989b]: On a problem of Pommerening in invariant theory. J. Algebra 126, 300–309MATHGoogle Scholar
  150. Humphreys, J.E. [1975]: Linear Algebraic Groups. New York: Springer. 269 pp. Zb1.325.20039Google Scholar
  151. Humphreys, J.E. [1978]: Hilbert’s fourteenth problem. Amer. Math. Monthly 85, 341–353MATHGoogle Scholar
  152. Igusa, J-i. [1970]: A classification of spinors up to dimension twelve. Am. J. Math. 92, 997–1028. Zb1.217,362MATHGoogle Scholar
  153. Igusa, J-i. [1973]: Geometry of absolutely admissible representations. In: Number Theory, Algebraic Geometry and Commutative Algebra. In Honour of Y. Akizuki, 373–552. Zb1.271.20022Google Scholar
  154. Ja’ja’, J. [1979]: An addendum to Kronecker’s theory of pencils. SIAM J. Appl. Math. 37, No. 3, 700–712. Zb1.425.15004Google Scholar
  155. Jósefiak, T., Pragacz, P., Weyman, J. [1981]: Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices. Astérisque 87/88, 109–189. Zb1.488.14012Google Scholar
  156. Kac, V.G. [1980]: Some remarks on nilpotent orbits. J. Algebra 64, 190–213. Zb1.431.17007MATHGoogle Scholar
  157. Kac, V.G. [1983]: Root systems, representations of quivers and invariant theory. Lect. Notes Math. 996, 75–108. Zb1.534.14004Google Scholar
  158. Kac, V.G., Popov, V.L., Vinberg, E.B. [1976]: Sur les groupes algébriques dont l’algèbre des invariants est libre. C.R. Acad. Sci., Paris, Sér. I 283, Sér. A, 875–878. Zb1.343.20023Google Scholar
  159. Kambayashi, T. [1966]: Projective representations of algebraic groups of transformations. Am. J. Math. 88, 199–205. Zb1.141,183MATHGoogle Scholar
  160. Kambayashi, T. [1979]: Automorphism group of a polynomial ring and algebraic group action on an affine space. J. Algebra 60, 439–451. Zb1.429.14017MATHGoogle Scholar
  161. Katanova, A.A. [1992]: Explicit form of certain multivector invariants. Adv. Sov. Math., vol. 8 [Vinberg, E.B., ed.]. American Math. Society, 87–94Google Scholar
  162. Kats, V.G., Kac, V.G. [1969]: Automorphisms of finite order of semisimple Lie algebras. Funkts. Anal. Prilozh. 3, No. 3, 94–96.Google Scholar
  163. Kats, V.G., Kac, V.G. [1969]: English transi.: Funct. Anal. Appl. 3, 252–254 [1970]. Zb1.274.17002Google Scholar
  164. Kats, V.G. [Kac, V.G.] [1975]: On the question of describing the orbit space of linear algebraic groups. Usp. Mat. Nauk 30, No. 6, 173–174 [Russian]. Zb1.391.20035MATHGoogle Scholar
  165. Katsylo, P.I. [1982]: Sections of sheets in a reductive Lie algebra. Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 3, 477–486.Google Scholar
  166. Katsylo, P.I. [1982]: English transi.: Math. USSR, Izv. 20, 449–458 [1983]. Zb1.533.17005Google Scholar
  167. Katsylo, P.I. [1983]: Rationality of orbit spaces of irreducible representations of SL2. Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 26–36.Google Scholar
  168. Katsylo, P.I. [1983]: English transi.: Math. USSR, Izv. 22, 23–32 [1984]. Zb1.569.14026Google Scholar
  169. Katsylo, P.I. [1984a]: Rationality of moduli spaces of hyperelliptic curves. Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 4, 705–710.Google Scholar
  170. Katsylo, P.I. [1984a]: English transl.: Math. USSR, Izv. 25, 45–50 [1985]. Zb1.593.14017Google Scholar
  171. Katsylo, P.I. [1984b]: Rationality of fields of invariants of reducible representations of SL2. Vestn. Mosk. Univ., Ser. I, No. 5, 77–79.Google Scholar
  172. Katsylo, P.I. [1984b]: English transi.: Mosc. Univ. Math. Bull. 39, No. 5, 80–83 [1984]. Zb1.582.20032MATHGoogle Scholar
  173. Katsylo, P.I. [1988]: Rationality of the moduli varieties of plane curves of degree 3k. Mat. Sb. Nov. Ser. 136, No. 3, 377–384 [Russian]Google Scholar
  174. Katsylo, P.I. [1990]: Stable rationality of the fields of invariants of linear representations of the groups PSL6 and PSL12. Mat. Zametki 48, No. 2, 49–52 [Russian]Google Scholar
  175. Katsylo, P.I. [1991]: Rationality of the moduli varieties of curves of genus 5. Mat. Sb. Nov. Ser. 182, No. 3, 457–464 [Russian]Google Scholar
  176. Katsylo, P.I. [1992]: On the birational geometry of the space of ternary quartics. Adv. Sov. Math., vol. 8 [Vinberg, E.B., ed.], American Mathematical Society, 95–104Google Scholar
  177. Katsylo, P.I. [1993]: Rationality of the moduli variety of curves of genus 3. Max Planck Institut. Preprint.Google Scholar
  178. Kawanaka, N. [1987]: Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra. J. Fac. Sci., Univ. Tokyo, Sect. I A 34, No. 3, 573–597. Zb1.651.20046MATHGoogle Scholar
  179. Kempf, G.R. [1978]: Instability in invariant theory. Ann. Math., II. Ser. 108, 299–316. Zb1.406.14031MATHGoogle Scholar
  180. Kempf, G.R. [1979]: The Hochster-Roberts theorem in invariant theory. Mich. Math. J. 26, 19–32.Zb1.4 09.13004MATHGoogle Scholar
  181. Kempf, G.R. [1980]: Some quotient surfaces are smooth. Mich. Math. J. 27, 295–299. Zb1.465.14018MATHGoogle Scholar
  182. Kempf, G.R. [1987]: Computing invariants. Lect. Notes Math. 1278, 81–94. Zb1.633.14007Google Scholar
  183. Kempf, G.R., Knudsen, F., Mumford, D., Saint-Donat, B. [1973]: Toroidal Embeddings. I. Lect. Notes Math. 339. 209 pp. Zb1.271.14017Google Scholar
  184. Kempf, G.R., Ness, L. [1979]: The length of vectors in representation spaces. Lect. Notes Math. 732, 233–243. Zb1.407.22012Google Scholar
  185. Kempken, G. [1982]: Induced conjugacy classes in classical Lie algebras. Prepr. 3, Math. Inst. Basel, 1–51, appeared in: Abh. Math. Semin. Univ. Hamb. 53, 53–83 [1983]. Zb1.495.17003Google Scholar
  186. Khadzhiev, D. [1967]: Some questions in the theory of vector invariants. Mat. Sb., Nov. Ser. 72, No. 3, 420–435.Google Scholar
  187. Khadzhiev, D. [1967]: English transi.: Math. USSR, Sb. 1, 383–396 [1967]. Zb1.166,297Google Scholar
  188. Khadzhiev, D. [1978]: Invariant Theory of Binary Forms. Tashkent: Branch of the Academy of Sciences. 52 pp. [Russian]. Zb1.465.15013Google Scholar
  189. Kimura, T. [1983]: A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications. J. Algebra 83, No. 1, 72–100. Zb1.533.14024MATHGoogle Scholar
  190. Kimura, T., Kasai, S-i. [1985]: The orbital decomposition of some prehomogeneous vector spaces. Adv. Stud. Pure Math. 6, 437–480. Zb1.577.14042Google Scholar
  191. Kimura, T., Kasai, S-i., Yasukura, O. [1986]: A classification of the representations of reductive algebraic groups which admit only a finite number of orbits. Am. J. Math. 108, No. 3, 643–691. Zb1.604.20044MATHGoogle Scholar
  192. Kirwan, F. [1984]: Cohomology of quotients in symplectic and algebraic geometry. Math. Notes. Princeton Univ. Press 31Google Scholar
  193. Klein, F. [1872]: Vergleichende Betrachtungen über Neuere Geometrische Forschungen. Erlangen. FdM4,229Google Scholar
  194. Klein, F. [1884]: Vorlesungen über des Ikosaeder und die Auflösung der Gleichungen vom Fünften Grade. Leipzig: Teubner. 260 pp. FdM 16,61Google Scholar
  195. Klein, F. [1925]: Elementarmathematik von Höheren Standpunkte aus. Band 2. Geometrie, 3 Aufl. Berlin: Springer. 314 pp. FdM40,538Google Scholar
  196. Klein, F. [1926]: Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert. Teil I. Berlin: Springer. 400 pp. FdM52,22Google Scholar
  197. Knop, F. [1986]: Über die Glattheit von Quotientabbildungen. Manuscr. Math. 56, 419–427. Zb1.585.14033MATHGoogle Scholar
  198. Knop, F. [1991]: Luna-Vust theory of spherical embeddings. In: Proc. Hyderabad Conf. on Algebraic Groups [Ramanan, S., ed.]. Manoj Prakashan, 225–250Google Scholar
  199. Knop, F. [1992]: Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins. Math. Inst. Basel. Preprint.Google Scholar
  200. Knop, F. [1993]: Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind. Math. Annalen 295, 333–363MATHGoogle Scholar
  201. Knop, F., Littelmann, P. [1987]: Der Grad erzeugender Funktionen von Invariantenringen. Math. Z. 196, 211–229. Zb1.635.20017MATHGoogle Scholar
  202. Kolmogorov, A.N., Yushkevich, A.P. [eds.] [1978]: Mathematics of the 19’s Century. Mathematical Logic, Algebra, Number Theory, Probability Theory. Moscow: Nauka. 255 pp. English transl.: Basel: Birkhäuser 1992. Zb1.616.00002.Google Scholar
  203. Kostant, B. [1963]: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404. Zb1.124,268MATHGoogle Scholar
  204. Kostant, B., Rallis, S. [1971]: Orbits and representations associated with symmetric spaces. Am. J. Math. 93, 753–809. Zb1.224.22013MATHGoogle Scholar
  205. Kraft, H. [1985]: Geometrische Methoden in der Invariantentheorie. Braunschweig/Wiesbaden: Vieweg. 308 pp. Zb1.569.14003Google Scholar
  206. Kraft, H., Petrie, T. Schwarz, G.W. [eds.] [1989]: Topological Methods in Algebraic Transformation Groups. Proceedings of the Conference Held at Rutgers University, New Brunswick, New Jersey, April 4–8, 1988. Prog. Math. 80. Birkhäuser. 210 pp. Zb1.683.00010Google Scholar
  207. Kraft, H., Popov, V.L. [1985]: Semisimple group actions on the three-dimensional affine space are linear. Comment. Math. HeIv. 60, 466–479. Zb1.645.14020MATHGoogle Scholar
  208. Kraft, H., Procesi, C. [1979]: Closures of conjugacy classes of matrices are normal. Invent. Math. 53, 227–247. Zb1.434.14026MATHGoogle Scholar
  209. Kraft, H., Procesi, C. [1981]: Minimal singularities in GL,,. Invent. Math. 62, 503–515. Zb1.478.14040MATHGoogle Scholar
  210. Kraft, H., Procesi, C. [1982]: On the geometry of conjugacy classes in classical groups. Comment. Math. HeIv. 57, 539–602. Zb1.511.14023MATHGoogle Scholar
  211. Kraft, H., Schwarz, G.W. [1992]: Reductive group actions with one-dimensional quotient. Publ. Math. IHES 76, 1–97MATHGoogle Scholar
  212. Kraft, H., Slodowy, P., Springer, T.A. [eds.] [1989]: Algebraische Transformationsgruppen und Invariantentheorie. DMV Semin., Band 13. Birkhäuser. 211 pp. Zb1.682.00008Google Scholar
  213. Krämer, M. [1979]: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math. 38, 129–153. Zb1.402.22006MATHGoogle Scholar
  214. Kung, J.P.S., Rota, G-C. [1984]: The invariant theory of binary forms. Bull. Am. Math. Soc., New Ser. 10, No. 1, 27–85. Zb1.577.15020MATHGoogle Scholar
  215. Kunz, E. [1985]: Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser. 238 pp. Zb1.432.13001Google Scholar
  216. Kuribayashi, A., Komiya, K. [1979]: On Weierstrass points and automorphisms of curves of genus three. Lect. Notes Math. 732, 253–299. Zb1.494.14012Google Scholar
  217. Lang, S. [1965]: Algebra. Reading, Mass.: Addison-Wesley. 508 pp. Zb1.193,347Google Scholar
  218. Lascoux, A. [1978]: Syzygies des variétés déterminantales. Adv. Math. 30, 202–237. Zb1.394.14022MATHGoogle Scholar
  219. Littelmann, P. [1989]: Koreguläre und äquidimensionale Darstellungen. J. Algebra 123, No. 1, 193–222. Zb1.688.14042MATHGoogle Scholar
  220. Littelmann, P. [1993]: On spherical double cones [to appear in J. Algebra]Google Scholar
  221. Luna, D. [1973]: Slices étales. Bull. Soc. Math. Fr., Suppl., Mém. 33, 81–105. Zb1.286.14014MATHGoogle Scholar
  222. Luna, D. [1975a]: Adhérences d’orbite et invariants. Invent. Math. 29, 231–238. Zb1.315.14018MATHGoogle Scholar
  223. Luna, D. [1975b]: Sur certaines opérations différentiables des groupes de Lie. Am. J. Math. 97, 172–181. Zb1.334.57022MATHGoogle Scholar
  224. Luna, D. [1993]: Sous-groupes sphériques résolubles. Prépubl. Inst. Fourier, No. 241Google Scholar
  225. Luna, D., Richardson, R.W. [1979]: A generalization of the Chevalley restriction theorem. Duke Math. J. 46, No. 3, 487–496. Zb1.444.14010MATHGoogle Scholar
  226. Luna, D., Vust, T. [1983]: Plongements d’espaces homogènes. Comment. Math. HeIv. 58, 186–245. Zb1.545.14010MATHGoogle Scholar
  227. Matsushima, Y. [1960]: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J. 16, 205–218. Zb1.94,282MATHGoogle Scholar
  228. Merzlyakov, Y.I. [1980]: Rational Groups. Moscow: Nauka. 464 pp. [Russian]. Zb1.518.20032Google Scholar
  229. Mikityuk, I.V. [1986]: On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Mat. Sb., Nov. Ser. 129, No. 4, 514–534.Google Scholar
  230. Mikityuk, I.V. [1986]: English transi.: Math. USSR, Sb. 57, 527–546 [1987]. Zb1.621.70005Google Scholar
  231. Milne, J.S. [1979]: Étale Cohomology. Princeton, N.J.: Princeton University Press. 323 pp. Zb1.433.14012Google Scholar
  232. Miyata, T. [1971]: Invariants of certain groups. I. Nagoya Math. J. 41, 69–73. Zb1.211,68MATHGoogle Scholar
  233. Molien, T. [1897]: Invarianten der linearen Substitutionsgruppen. Sitzungsber. Königl. Preuss. Akad. Wiss., 1152–1156. FdM28,115Google Scholar
  234. Muller, I., Rubenthaler, H., Schiffmann, G. [1986]: Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées. Math. Ann. 274, No. 1, 95–123. Zb1.568.17007MATHGoogle Scholar
  235. Mumford, D. [1961]: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math., Inst. Hautes Étud. Sci. 9, 5–22. Zb1.108,168MATHGoogle Scholar
  236. Mumford, D. [1965]: Geometric Invariant Theory. Berlin: Springer. 147 pp. Zb1.147,393Google Scholar
  237. Mumford, D. [1977]: Stability of projective varieties. Enseign. Math., II. Ser. 24, 39–110. Zb1.363.14003Google Scholar
  238. Mumford, D., Fogarty, J. [1982]: Geometric Invariant Theory. Berlin: Springer. 219 pp. Zb1.504.14008MATHGoogle Scholar
  239. Murthy, M. [1964]: A note on factorial rings. Arch. Math. 15, 418–420. Zb1.123,34MATHGoogle Scholar
  240. Nagata, M. [1956]: On the embedding problem of abstract varieties in projective varieties. Mem. Kyoto Univ. 30, 71–82. Zb1.70,160MATHGoogle Scholar
  241. Nagata, M. [1959]: On the 14`h problem of Hilbert. Am. J. Math. 81, 766–772. Zb1.192,138MATHGoogle Scholar
  242. Nagata, M. [1964]: Invariants of a group in an affine ring. J. Math. Kyoto Univ. 3, 369–377. Zb1.146,45MATHGoogle Scholar
  243. Nagata, M. [1965]: Lectures on the Fourteenth Problem of Hilbert. Bombay: Tata Institute of Fundamental Research. 78 pp. Zb1.182,541Google Scholar
  244. Nakajima, H. [1979]: Invariants of finite groups generated by pseudoreflections in positive characteristic. Tsukuba J. Math. 3, No. 1, 109–122. Zb1.418.20041MATHGoogle Scholar
  245. Nakajima, H. [1980]: Modular representations of p-groups with regular rings of invariants. Proc. Japan Acad., Ser. A 56, No. 10, 469–473. Zb1.481.20011MATHGoogle Scholar
  246. Nakajima, H. [1982]: Modular representation of abelian groups with regular rings of invariants. Nagoya Math. J. 86, 229–248. ZbI.443.14005MATHGoogle Scholar
  247. Nakajima, H. [1984]: Quotient singularities which are complete intersections. Manuscr. Math. 48, 163–187. Zb1.577.14038MATHGoogle Scholar
  248. Nakajima, H. [1985]: Quotient complete intersections of affine spaces by finite linear groups. Nagoya Math. J. 98, 1–36. Zb1.596.14038MATHGoogle Scholar
  249. Nakajima, H. [1986]: Representations of a reductive algebraic group whose algebras of invariants are complete intersections. J. Reine Angew. Math. 367, 115–138. Zb1.575.20036MATHGoogle Scholar
  250. Nakajima, H., Watanabe, K-i. [1984]: The classification of quotient singularities which are complete intersections. Lect. Notes Math. 1092, 102–120. Zb1.577.14039Google Scholar
  251. Ness, L. [1979]: Mumford’s numerical function and stable projective hypersurfaces. Lect. Notes Math. 732, 417–453. Zb1.442.14019Google Scholar
  252. Ness, L. [1984]: A stratification of the null cone via the moment map. Am. J. Math. 106, 1281–1329. Zb1.604.14006MATHGoogle Scholar
  253. Newstead, P.E. [1978]: Introduction to Moduli Problems and Orbit Spaces. Bombay: Tata Institute of Fundamental Research. 183 pp., published by Berlin: Springer. Zb1.411.14003Google Scholar
  254. Newstead, P.E. [1982]: Covariants of pencils of binary cubics. Proc. R. Soc. Edinb., Sect. A 91, 181–183. Zb1.505.14004Google Scholar
  255. Noether, E. [1915a]: Körper und Systeme rationaler Funktionen. Math. Ann. 76, 161–196. FdM46,1442MATHGoogle Scholar
  256. Noether, E. [1915b]: Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77, 89–92. FdM45,198MATHGoogle Scholar
  257. Onishchik, A.L. [1960]: Complex hulls of compact homogeneous spaces. Dokl. Akad. Nauk SSSR 130, No. 4, 726–729.Google Scholar
  258. Onishchik, A.L. [1960]: English transi.: Sov. Math., Dokl. 1, 88–91 [1960]. Zb1.90,94Google Scholar
  259. Panyushev, D.I. [1982a]: Classification of four-dimensional anticommutative algebras with nontrivial group of automorphisme. In: Vopr. Teor. Grupp. Gomologicheskoj Algebry 1982, 98106. Zbl.567.20027Google Scholar
  260. Panyushev, D.I. [1982b]: On the orbit spaces of finite and connected linear groups. Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 1, 95–99.Google Scholar
  261. Panyushev, D.I. [1982b]: English transi.: Math. USSR, Izv. 20, 97–101 [1983]. Zb1.517.20019Google Scholar
  262. Panyushev, D.I. [1983]: Semisimple automorphism groups of a four-dimensional affine space. Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 4, 881–894.Google Scholar
  263. Panyushev, D.I. [1983]: English transi.: Math. USSR, Izv. 23, 171–183 [1984]. Zb1.581.14033Google Scholar
  264. Panyushev, D.I. [1984]: Regular elements in spaces of linear representations of reductive algebraic groups. Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 2, 411–419.Google Scholar
  265. Panyushev, D.I. [1984]: English transi.: Math. USSR, Izv. 24, 383–390 [1985]. Zbl.557.20026Google Scholar
  266. Panyushev, D.I. [1985]: Regular elements in spaces of linear representations. II. Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 5, 979–985.Google Scholar
  267. Panyushev, D.I. [1985]: English transi.: Math. USSR, Izv. 27, 279–284 [1986]. Zb1.618.20029Google Scholar
  268. Panyushev, D.I. [1987]: Orbits of maximal dimension of solvable subgroups of reductive linear groups and reduction for U-invariants. Mat. Sb. Nov. Ser. 132, No. 3, 371–382.Google Scholar
  269. Panyushev, D.I. [1987]: English transi.: Math. USSR, Sb. 60, No. 2, 365–375 [1988]. Zb1.628.20034MATHGoogle Scholar
  270. Panyushev, D.I. [1990]: Complexity and rank of homogeneous spaces. Geom. Dedic. 34, 249–269MATHGoogle Scholar
  271. Panyushev, D.I. [1991]: Rationality of the singularities and Gorensteinness of nilpotent orbits. Funkts. Anal. Prilozh. 25, No. 3, 76–78Google Scholar
  272. Panyushev, D.I. [1992]: Complexity of quasiaffine homogeneous varieties, t-decompositions and affine homogeneous spaces of complexity 1. Adv. Sov. Math., vol. 8 [Vinberg, E.B., ed.]. American Math. Society, 151–166Google Scholar
  273. Panyushev, D.I. [1993a]: On semisimple groups, admitting a finite coregular extension. Funkts. Anal. Prilozh. 27, No. 2 [Russian]Google Scholar
  274. Panyushev, D.I. [1993b]: Complexity and rank of double cones and tensor product decompositions. Comm. Math. HeIv. 68 [to appear]Google Scholar
  275. Panyushev, D.I. [1993e]: Restriction theorem and Poincaré series for U-invariants. Max-PlanckInstitut. Preprint 93/10Google Scholar
  276. Panyushev, D.I. [1993d]: Complexity and nilpotent orbits. Max-Planck-Institut. Preprint 93/43Google Scholar
  277. Parshin, A.N. [1974]: D. Hilbert and invariant theory. Istor: Mat. Issled. 20, 171–197. [Russian]. Zb1.372.01003Google Scholar
  278. Pauer, F. [1981]: Normale Einbettungen von G/U. Math. Ann. 257, 371–396. Zb1.461.14013MATHGoogle Scholar
  279. Pauer, F. [1983]: Glatte Einbettungen von G/U. Math. Ann. 262, 421–429. Zb1.512.14029MATHGoogle Scholar
  280. Peterson, D. [1978]: Geometry of the Adjoint Representation of a Complex Semisimple Lie Algebra. Ph.D. Thesis, Harvard UniversityGoogle Scholar
  281. Poincaré, H. [1881]: Sur les formes cubiques ternaires et quaternaires. I. J. Éc. Polytech. 50, 199–253. FdM15,97Google Scholar
  282. Poincaré, H. [1882]: Sur les formes cubiques ternaires et quaternaires. II. J. L. Polytech. 51, 45–91. FdM 15,97Google Scholar
  283. Pommerening, K. [1981]: Invarianten unipotenter Gruppen. Math. Z. 176, 359–374 Pommerening, K. [1987a]: Invariants of unipotent groups [a survey]. Lecture Notes in Math. 1278, Springer-Verlag, 8–17MATHGoogle Scholar
  284. Pommerening, K. [1987b]: Ordered sets with the standardizing property and straightening laws for algebras of invariants. Adv. Math. 63, 271–290MATHGoogle Scholar
  285. Popov, A.M. [1985]: Finite isotropy subgroups in general position in simple linear Lie groups. Tr. Mosk. Mat. O.-va 48, 7–59.Google Scholar
  286. Popov, A.M. [1985]: English transi.: Trans. Mosc. Math. Soc. 1986, 3–63 [1988]. Zb1.653.22007Google Scholar
  287. Popov, A.M. [1987]: Finite isotropy subgroups in general position in irreducible semisimple linear Lie groups. Tr. Mosk. Mat. 0.-va 50, 209–248.MATHGoogle Scholar
  288. Popov, A.M. [1987]: English trans].: Trans. Mosc. Math. Soc. 1988, 205–249 [1988]. Zb1.661.22009MATHGoogle Scholar
  289. Popov, V.L. [1970]: A stability criterion for an action of a semisimple group on a factorial variety. Izv. Akad. Nauk SSSR, Ser. Mat. 34, No. 3, 523–531.MATHGoogle Scholar
  290. Popov, V.L. [1970]: English transi.: Math. USSR, Izv. 4, 527–535 [1971]. Zb1.261.14011Google Scholar
  291. Popov, V.L. [1972]: On the stability of the action of an algebraic group on an algebraic variety. Izv. Akad. Nauk SSSR, Ser. Mat. 36, No. 2, 371–385.MATHGoogle Scholar
  292. Popov, V.L. [1972]: English trans].: Math. USSR, Izv. 6, 367–379 [1973]. Zb1.232.14018Google Scholar
  293. Popov, V.L. [1973]: Quasihomogeneous affine algebraic varieties of the group SL2. Izv. Akad. Nauk SSSR, Ser. Mat. 37, No. 4, 792–832.MATHGoogle Scholar
  294. Popov, V.L. [1973]: English transi.: Math. USSR, Izv. 7, 793–831 [1974]. Zb1.281.14022Google Scholar
  295. Popov, V.L. [1974]: The Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles. Izv. Akad. Nauk SSSR, Ser. Mat. 38, No. 2, 292–322.Google Scholar
  296. Popov, V.L. [1974]: English transi.: Math. USSR, Izv. 8, 301–327 [1975]. Zb1.298.140023Google Scholar
  297. Popov, V.L. [1976]: Representations with a free module of covariants. Funkts. Anal. Prilozh. 10, No. 3, 91–92.Google Scholar
  298. Popov, V.L. [1976]: English trans].: Funct. Anal. Appl. 10, 242–244 [1977]. Zb1.365.20053Google Scholar
  299. Popov, V.L. [1978a]: Algebraic curves with an infinite group of automorphisms. Mat. Zametki 23, No. 2, 183–195.MATHGoogle Scholar
  300. Popov, V.L. [1978a]: English transi.: Math. Notes. 23, 102–108 [1978]. Zb1.382.14017MATHGoogle Scholar
  301. Popov, V.L. [1978b]: Classification of spinors of dimension fourteen. Tr. Mosk. Mat. 0.-va 37, 173–217.Google Scholar
  302. Popov, V.L. [1978b]: English trans].: Trans. Mosc. Math. Soc. 1, 181–232 [1980]. Zb1.432.20036MATHGoogle Scholar
  303. Popov, V.L. [1979a]: Invariant theory. In: Mathematics Encyclopedia, Vol. 2, 540–543. Moscow: Soviet Encyclopedia [Russian]Google Scholar
  304. Popov, V.L. [1979b]: On Hilbert’s theorem on invariants. Dokl. Akad. Nauk SSSR 249, No. 3, 551–555.Google Scholar
  305. Popov, V.L. [1979b]: English transi.: Sov. Math. Dokl. 20, 1318–1322 [1979]. Zb1.446.14004MATHGoogle Scholar
  306. Popov, V.L. [1981a]: Constructive invariant theory. Izv. Akad. Nauk SSSR, Ser. Mat. 45, No. 5, 1100–1120.MATHGoogle Scholar
  307. Popov, V.L. [1981a]: English transi.: Math. USSR, Izv. 19, 359–376 [1982]. Zb1.478.14006Google Scholar
  308. Popov, V.L. [1981b]: Constructive invariant theory. Astérisque 87/88, 303–334. Zb1.491.14004Google Scholar
  309. Popov, V.L. [1982]: A finiteness theorem for representations with a free algebra of invariants. Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 347–370.Google Scholar
  310. Popov, V.L. [1982]: English transi.: Math. USSR, Izv. 20, 333–354 [1988]. Zb1.547.20034Google Scholar
  311. Popov, V.L. [1983a]: Syzygies in invariant theory. Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 3, 544–622.Google Scholar
  312. Popov, V.L. [1983a]: English trans].: Math. USSR, Izv. 22, 507–585 [1984]. Zb1.573.14003Google Scholar
  313. Popov, V.L. [1983b]: Homological dimension of algebras of invariants. J. Reine Angew. Math. 341, 157–173. Zb1.525.14007MATHGoogle Scholar
  314. Popov, V.L. [1986a]: Contraction of actions of reductive algebraic groups. Mat. Sb., Nov. Ser. 130, No. 3, 310–334.Google Scholar
  315. Popov, V.L. [1986a]: English transi.: Math. USSR Sb. 58, 311–335 [1987]. Zb1.613.14034MATHGoogle Scholar
  316. Popov, V.L. [1986b]: Modern developments in invariant theory. In: Proc. Int. Congr. Math. Berkeley Calif. 1986, Vol. 1 394–406. Zb1.679.14024Google Scholar
  317. Popov, V.L. [1987]: One hundred fifty years of invariant theory. Metodol. Anal. Mat. Teor., Collect. Sci. Works, Moskva, 235–256 [Russian]. Zb1.721.01007Google Scholar
  318. Popov, V.L. [1988]: Closed orbits of Borel subgroups. Mat. Sb., Nov. Ser. 135, No. 3, 385–402. English transi.: Math. USSR, Sb. 63, 375–392 [1989]. Zb1.713.20036Google Scholar
  319. Popov, V.L. [1991]: Invariant theory. In: First Siberian Winter School “Algebra and Analysis”. Transi., II. Ser., Am. Math. Soc. 148, 99–112. Zb1.736.15019Google Scholar
  320. Popov, V.L. [1992a]: Groups, Generators, Syzygies and Orbits in Invariant Theory. Transl. Math. Monogr. 100. Providence, R.I.: American Mathematical Society. 245 pp.Google Scholar
  321. Popov, V.L. [1992b]: On the “Lemma of Seshadri”. Adv. Sov. Math., vol. 8 [Vinberg E.B., ed.], American Mathematical Society, 167–172Google Scholar
  322. Popov, V.L. [1993]: Lectures on invariant theory [written with assistance of S. Grimm and A. Kurth]. To appear in Lectures in Mathematics-ETH ZürichGoogle Scholar
  323. Popp, H. [1977]: Moduli Theory and Classification Theory of Algebraic Varieties. Lect. Notes Math. 620, 189 pp. Zb1.359.14005Google Scholar
  324. Premet, A.A. [1991]: The theorem on restriction of invariants, and nilpotent elements in W„. Mat. Sb. 182, No. 5, 746–773.MATHGoogle Scholar
  325. Premet, A.A. [1991]: English transi.: Math. USSR, Sb. 73, No. 1, 135–159 [1992]. Zb1.737.17006Google Scholar
  326. Procesi, C. [1976]: The invariant theory of n x n matrices. Adv. Math. 19, 306–381. Zb1.331.15021MATHGoogle Scholar
  327. Procesi, C. [1982]: A Primer of Invariant Theory. Brandeis Lect. Notes 1. 218 pp.Google Scholar
  328. Procesi, C., Schwarz, G. [1985]: Inequalities defining orbit spaces. Invent. Math. 81, 539–554. Zb1.578.14010MATHGoogle Scholar
  329. Pyasetskij, V.S. [1975]: Linear Lie groups acting with a finite number of orbits. Funkts. Anal. Prilozh. 9, No. 4, 85–86.Google Scholar
  330. Pyasetskij, V.S. [1975]: English transi.: Funct. Anal. Appl. 9, 351–353 [1976]. Zb1.326.22004Google Scholar
  331. Pyasetskij, V.S. [1983]: Estimate of the number of orbits of a triangular group. In: Vopr. Teor. Grupp. Gomologicheskoj Algebry 1983, 54–57. Zb1.599.20061Google Scholar
  332. Pyasetskij, V.S. [1985]: Fundamental sets of triangular groups. In: Vopr. Teor. Grupp. Gomologicheskoj Algebry 1985, 157–160. Zb1.734.14020Google Scholar
  333. Razmyslov, Y.P. [1974]: Trace identities of full matrix algebras over a field of characteristic zero. Izv. Akad. Nauk SSSR, Ser. Mat. 38, No. 4, 723–756.Google Scholar
  334. Razmyslov, Y.P. [1974]: English transi.: Math. USSR, Izv. 8, 727–760 [1975]. Zb1.311.16016Google Scholar
  335. Richardson, R.W. [1967]: Conjugacy classes in Lie algebras and algebraic groups. Ann. Math., II. Ser. 86, 1–15. Zb1.153,45Google Scholar
  336. Richardson, R.W. [1972a]: Principal orbit types for algebraic transformation spaces in characteristic zero. Invent. Math. 16, 6–14. Zb1.242.14010MATHGoogle Scholar
  337. Richardson, R.W. [1972b]: Deformations of Lie subgroups and the variation of isotropy subgroups. Acta Math. 129, 35–73. Zb1.242.22020MATHGoogle Scholar
  338. Richardson, R.W. [1977]: Affine coset spaces of reductive algebraic groups. Bull. Lond. Math. Soc. 9, 38–41. Zb1.355.14020MATHGoogle Scholar
  339. Richardson, R.W. [1979]: The conjugating representation of a semisimple group. Invent. Math. 54, No. 3, 229–245MATHGoogle Scholar
  340. Richmond, D.R. [1990]: On vector invariants over finite fields. Adv. in Math. 81, 30–65Google Scholar
  341. Rivlin, R.S., Smith, G.F. [1969]: Orthogonal integrity basis for N symmetric matrices. In: Contributions to Mechanics. 121–141. Pergamon PressGoogle Scholar
  342. Roberts, P. [1990]: An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem. J. Algebra 132, 461–473. Zb1.716.13013MATHGoogle Scholar
  343. Rosenlicht, M. [1956]: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443. Zb1.73,376MATHGoogle Scholar
  344. Rosenlicht, M. [1961]: Toroidal algebraic groups. Proc. Am. Math. Soc. 12, 984–988. Zb1.107,147MATHGoogle Scholar
  345. Rousseau, G. [1978]: Immeubles sphériques et théorie des invariants. C.R. Acad. Sci., Paris, Sér. 1 286, 247–250. Zb1.375.14013Google Scholar
  346. Saltman, D. [1984]: Noether’s problem over an algebraically closed field. Invent. Math. 77, 71–84. Zbl.546.14014MATHGoogle Scholar
  347. Saltman, D. [1985]: Groups acting on fields: Noether’s problem. Contemp. Math. 43, 267–277. Zb1.568.12013Google Scholar
  348. Sampson, J.H., Washnitzer, G. [1959]: A Künneth formula for coherent algebraic sheaves. Ill. J. Math. 3, 389–402. Zb1.88,394MATHGoogle Scholar
  349. Sato, M., Kimura, T. [1974]: On zeta functions associated with prehomogeneous vector spaces. Ann. Math., II. Ser. 100, 131–170. Zb1.309.10014MATHGoogle Scholar
  350. Sato, M., Kimura, T. [1977]: Classification of irreducible prehomogeneous vector spaces and their invariants. Nagoya Math. J. 65, 1–155. Zb1.321.14030MATHGoogle Scholar
  351. Schmid, B. [1991]: Finite groups and invariant theory. Lecture Notes Math. 1478, 35–66Google Scholar
  352. Schoefield, A.H. [1991]: Semi-invariants of quivers. J. London Math. Soc. 43, No. 3, 385–395Google Scholar
  353. Schoefield, A.H. [1992]: General representations of quivers. Proc. London Math. Soc. 65, No. 1, 46–64Google Scholar
  354. Schur, I. [1968]: Vorlesungen über Invariantentheorie. Berlin: Springer. 134 pp. Zb1.159,37Google Scholar
  355. Schwarz, G.W. [1978a]: Representations of simple Lie groups with regular rings of invariants. Invent. Math. 49, 167–191. Zb1.391.20032MATHGoogle Scholar
  356. Schwarz, G.W. [1978b]: Representations of simple Lie groups with a free module of covariants. Invent. Math. 50, 1–12. Zb1.391.20033Google Scholar
  357. Schwarz, G.W. [1983]: Invariant theory of G 2. Bull. Am. Math. Soc. 9, 335–338. Zb1.531.14007MATHGoogle Scholar
  358. Schwarz, G.W. [1987]: On classical invariant theory and binary cubics. Ann. Inst. Fourier 37, No. 3, 191–216. Zb1.597.14011MATHGoogle Scholar
  359. Schwarz, G.W. [1988]: Invariant theory of G2 and Spin,. Comment. Math. Helv. 63, No. 4, 624–663. Zb1.664.14006MATHGoogle Scholar
  360. Schwarz, G.W. [1989]: Exotic algebraic group actions. C.R. Acad. Sci., Paris, Sér. I 309, 89–94. Zb1.688.14040Google Scholar
  361. Serre, J-P. [1964]: Cohomologie Galoisienne. Lect. Notes Math. 5. 219 pp. Zb1.128,263Google Scholar
  362. Serre, J-P. [1975]: Algèbre Locale. Multiplicités. Lect. Notes Math. 11. 160 pp. Zb1.296.13018Google Scholar
  363. Servedio, F. J. [1973]: Prehomogeneous vector spaces and varieties. Trans. Am. Math. Soc. 176, 421–444. Zb1.266.20043MATHGoogle Scholar
  364. Seshadri, C.S. [1962]: On a theorem of Weitzenböck in invariant theory. J. Math. Kyoto Univ. 1, 403–409. Zb1.112,254MATHGoogle Scholar
  365. Seshadri, C.S. [1963]: Some results on the quotient space by an algebraic group of automorphisms. Math. Ann. 149, 286–301. Zb1.113,363MATHGoogle Scholar
  366. Seshadri, C.S. [1972]: Quotient spaces modulo reductive algebraic groups. Ann. Math., II. Ser. 95, 511–556. Zb1.241.14024MATHGoogle Scholar
  367. Seshadri, C.S. [1977]: Geometric reductivity over an arbitrary base. Adv. Math. 26, 225–274. Zb1.371.14009MATHGoogle Scholar
  368. Shafarevich, I. [1966]: Some infinite-dimensional algebraic groups. Rend. Mat. Appl., V. Ser. 25, 208–212. Zb1.149,390Google Scholar
  369. Shafarevich, I.R. [1988]: Foundations of Algebraic Geometry. Vols. 1, 2. Moscow: Nauka. 351 pp., 304 pp. Zb1.675.14001. English transl. of the first ed. [1972, Zb1.258.14001]: Berlin: Springer 1977.Google Scholar
  370. Shephard, G.C., Todd, J.A. [1954]: Finite unitary reflection groups. Can. J. Math. 6, 274–304. Zb1.55,143MATHGoogle Scholar
  371. Shepherd-Barron, N.I. [1987]: The rationality of certain spaces associated to trigonal curves. Proc. Symp. Pure Math. 46, 165–171. Zb1.669.14015Google Scholar
  372. Shepherd-Barron, N.I. [1988]: The rationality of some moduli spaces of plane curves. Compos. Math. 67, 51–88. Zb1.661.14022MATHGoogle Scholar
  373. Shepherd-Barron, N.I. [1989]:-Invariant theory for S 5 and the rationality of M6. Comp. Math. 70, 13–25Google Scholar
  374. Shioda, T. [1967]: On the graded ring of invariants of binary octavics. Am. J. Math. 89, 1022–1046. Zb1.188,533MATHGoogle Scholar
  375. Shpiz, G.B. [1978]: Classification of irreducible locally transitive linear Lie groups. In: Geometric Methods in Problems of Algebra and Analysis, Interuniv. Thematic Work Collect., Yaroslavl 1978, 152–160 [Russian]. Zb1.415.22015Google Scholar
  376. Sibirskij, K.S. [1976]: Algebraic Invariants of Differential Equations and Matrices. Kishinev: Shtiintsa. 268 pp. [Russian]. Zb1.334.34014Google Scholar
  377. Sibirskij, K.S. [1982]: Introduction to the Algebraic Theory of Invariants of Differential Equations. Kishinev: Shtiintsa. 166 pp. English transi.: Manchester Univ. Press. 1988. Zb1.559.34046Google Scholar
  378. Simoniya, V.T. [1960]: The first fundamental theorem in the theory of vector invariants of the exceptional Lie group G2. Soobshch. Akad. Nauk Gruz. SSR 24, No. 6, 641–648 [Russian]. Zb1.93,253Google Scholar
  379. Slodowy, P. [1980]: Simple Singularities and Simple Algebraic Groups. Lect. Notes Math. 815. 175 pp. Zb1.441.14002Google Scholar
  380. Smith, G.F. [1970]: A complete set of unitary invariants for N 3 x 3 complex matrices. Tensor, New Ser. 21, 273–283. Zb1.202,37MATHGoogle Scholar
  381. Smoke, W. [1972]: Dimension and multiplicity for graded algebras. J. Algebra 21, 149–173. Zb1.231.13006MATHGoogle Scholar
  382. Spaltenstein, N. [1982]: Classes Unipotentes et Sous-Groupes de Borel. Lect. Notes Math. 946. 259 pp. Zb1.486.20025Google Scholar
  383. Spaltenstein, N. [1983]: Dégénérescences des formes bilinéaires. J. Algebra 80, 1–28. Zb1.505.15011MATHGoogle Scholar
  384. Spencer, A.J.M. [1971]: Theory of Invariants. New York, London. 125 pp. Zb1.339.15012Google Scholar
  385. Springer, T. [1976]: Trigonometric sums, Green functions of finite groups and representations of Weyl sums. Invent. Math. 36, 173–207. Zb1.374.20054MATHGoogle Scholar
  386. Springer, T. [1977]: Invariant Theory. Lect. Notes Math. 585. 111 pp. Zb1.346.20020Google Scholar
  387. Stanley, R. [1978]: Hilbert functions of graded algebras. Adv. Math. 28, 57–83. Zb1.384.13012MATHGoogle Scholar
  388. Stanley, R. [1979]: Invariants of finite groups and their applications to combinatorics. Bull. Am. Math. Soc., New. Ser. 1, 475–511. Zb1.497.20002MATHGoogle Scholar
  389. Steinberg, R. [1974]: Conjugacy Classes in Algebraic Groups. Lect. Notes Math. 366. 159 pp. Zb1.281.20037Google Scholar
  390. Sukhanov, A.A. [1978]: A description of observable subgroups of linear algebraic groups. Usp. Mat. Nauk 33, No. 2, 182–183 [Russian]Google Scholar
  391. Sukhanov, A.A. [1988]: A description of observable subgroups of linear algebraic groups. Mat. Sb., Nov. Ser. 137, No. 1, 90–102.Google Scholar
  392. Sukhanov, A.A. [1988]: English transi.: Math. USSR, Sb. 65, No. 1, 97–108 [1990]. Zb1.663.20043MATHGoogle Scholar
  393. Sumihiro, H. [1974]: Equivariant completion. J. Math. Kyoto Univ. 14, No. 1, 1–28. Zb1.277.14008MATHGoogle Scholar
  394. Tan, L. [1988]: On the Popov—Pommerening conjecture for the groups of type G2. Algebra, Groups and Geometries 5, 421–432MATHGoogle Scholar
  395. Tan, L. [1989a]: On the Popov—Pommerening conjecture for the groups of type A„. Proc. Amer. Math. Soc. 106, 611–616MATHGoogle Scholar
  396. Tan, L. [19896]: Some recent developments in the Popov—Pommerening conjecture. In: Group Actions, and Invariant Theory, CMS Conf. Proc. 10, Amer. Math. Soc., Providence, RI, 207220Google Scholar
  397. Tan, L. [1992]: An elementary counterexample to Hilbert’s fourteenth problem. Prepr., West Chester UniversityGoogle Scholar
  398. Thrall, R.M., Chanler, J.H. [1938]: Ternary trilinear forms in the field of complex numbers. Duke Math. J. 4, No. 4, 678–690. Zb1.20,61Google Scholar
  399. Trautman, K. [1992]: Orbits that always have affine stable neighbourhoods, Adv. Math. 91, No. 1, 54–63MATHGoogle Scholar
  400. Triantaphyllou, D. [1980]: Invariants of finite groups acting nonlinearly on rational function fields. J. Pure Appl. Algebra 18, 315–331. Zb1.471.12018MATHGoogle Scholar
  401. Van der Kulk, W. [1953]: On polynomial rings in two variables. Nieuw Arch. Wiskd. 1, 33–41. Zb1.50,260MATHGoogle Scholar
  402. Varchenko, A.N. [1972]: Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings. Izv. Akad. Nauk SSSR, Ser. Mat. 36, No. 5, 957–1019.Google Scholar
  403. Varchenko, A.N. [1972]: English transi.: Math. USSR, Izv. 6, 949–1008 [1973]. Zb1.251.10046Google Scholar
  404. Veles’ko, A.E. [1985]: A criterion of freeness of the rings of invariants of quadratic groups. Dokl. Akad. Nauk BSSR 29, no. 10, 869–871MATHGoogle Scholar
  405. Veles’ko, A.E. [1986]: The existence of groups generated by reflections, with infinitely generated rings of invariants. Dokl. Akad. Nauk. BSSR 30, no. 2, 105–107MATHGoogle Scholar
  406. Vinberg, E.B. [1960]: Invariant linear connections in a homogeneous space. Tr. Mosk. Mat. 0.-va 9, 191–210 [Russian]. Zb1.113,158Google Scholar
  407. Vinberg, E.B. [1975a]: On linear groups connected with periodic automorphisms of semisimple algebraic groups. Dokl. Akad. Nauk SSSR 221, No. 4, 767–770.Google Scholar
  408. Vinberg, E.B. [1975a]: English transi.: Soy. Math., Dokl. 16, 406–409 [1975]. Zb1.334.20020Google Scholar
  409. Vinberg, E.B. [1975b]: On the classification of nilpotent elements of graded Lie algebras. Dokl. Akad. Nauk SSSR 225, No. 4, 745–748.Google Scholar
  410. Vinberg, E.B. [1975b]: English transi.: Sov. Math., Dokl. 16, 1517–1520 [1976]. Zb1.374.17001Google Scholar
  411. Vinberg, E.B. [1976]: The Weyl group of a graded Lie algebra. Izv. Akad. Nauk SSSR, Ser. Mat. 40, No. 3, 488–526.MATHGoogle Scholar
  412. Vinberg, E.B. [1976]: English transi.: Math. USSR, Izv. 10, 463–495 [1977]. Zb1.363.20035Google Scholar
  413. Vinberg, E.B. [1979]: A classification of homogeneous nilpotent elements of a semisimple graded Lie algebra. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 19, 155–177.Google Scholar
  414. Vinberg, E.B. [1979]: English transi.: Sel. Math. Sov. 6, 15–35 [1987]. Zb1.431.17006Google Scholar
  415. Vinberg, E.B. [1980]: On the closure of an orbit of a reductive linear group. In: Algebra, Collect. Works, Moskva 1980, 31–36 [Russian]. Zb1.475.20032Google Scholar
  416. Vinberg, E.B. [1982a]: Rationality of the field of invariants of a triangular group. Vestn. Mosk. Univ., Ser. I, No. 2, 23–24.Google Scholar
  417. Vinberg, E.B. [1982a]: English transi.: Mosc. Univ. Math. Bull. 37, No. 2, 27–29 [1992]. Zb1.493.14005Google Scholar
  418. Vinberg, E.B. [1982b]: Effective invariant theory. In: Algebra, Collect. Works, Moskva 1982, 27–33.Google Scholar
  419. Vinberg, E.B. [1982b]: English transi.: Transi., II. Ser., Am. Math. Soc. 137, 15–19 [1987]. Zb1.538.15012Google Scholar
  420. Vinberg, E.B. [1986]: Complexity of actions of reductive groups. Funkts. Anal. Prilozh. 20, No. 1, 1–13.MATHGoogle Scholar
  421. Vinberg, E.B. [1986]: English transi.: Funct. Anal. Appl. 20, 1–11 [1986]. Zb1.601.14038Google Scholar
  422. Vinberg, E.B., Ehlashvili, A.G. [1978]: A classification of trivectors of a nine-dimensional space. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 18, 197–233.Google Scholar
  423. Vinberg, E.B., Ehlashvili, A.G. [1978]: English transi.: Sel. Math. Sov. 7, No. 1, 63–98 [1988]. Zb1.441.15015MATHGoogle Scholar
  424. Vinberg, E.B., Gorbatsevich, V.V., Onishchik, A.L. [1990]: Structure of Lie groups and Lie algebras. Itogi Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental’nye Napravleniya, vol. 41. VINITI, Moscow, 258 pp.; English translation in Encyclopaedia of Math. Sci., vol. 41, Springer-Verlag, 1994Google Scholar
  425. Vinberg, E.B., Kimel’fel’d, B.N. [1978]: Homogeneous domains on flag varieties and spherical subgroups of semisimple Lie groups. Funkts. Anal. Prilozh. 12, No. 3, 12–19.Google Scholar
  426. Vinberg, E.B., Kimel’fel’d, B.N. [1978]: English transi.: Funct. Anal. Appl. 12, 168–174 [1979]. Zb1.439.53055Google Scholar
  427. Vinberg, E.B., Onishchik, A.L. [1988]: Seminar on Lie Groups and Algebraic Groups. Moscow: Nauka. 343 pp. English transi.: Berlin: Springer 1990. Zbl. 648.22009, Zb1.722.22004Google Scholar
  428. Vinberg, E.B., Popov, V.L. [1972]: On a class of quasihomogeneous affine varieties. Izv. Akad. Nauk SSSR, Ser. Mat. 36, No. 4, 749–764.MATHGoogle Scholar
  429. Vinberg, E.B., Popov, V.L. [1972]: English transi.: Math. USSR, Izv. 6, 743–758 [1973]. Zb1.248.14014Google Scholar
  430. Voskresenskij, V.E. [1977]: Algebraic Tori. Moscow: Nauka. 223 pp. [Russian]. Zb1.499.14013Google Scholar
  431. Vust, T. [1976]: Sur la théorie des invariants des groupes classiques. Ann. Inst. Fourier 26, No. 1, 1–31. Zb1.314.20035MATHGoogle Scholar
  432. Vust, T. [1977]: Sur la théorie classique des invariants. Comment. Math. Heiv. 52, 259–295. Zb1.364.15022MATHGoogle Scholar
  433. Vust, T. [1980]: Foncteurs polynomiaux et théorie des invariants. Lect. Notes Math. 795, 330–340. Zb1.429.14003Google Scholar
  434. Vust, T. [1987]: Plongements d’espaces symétriques algébriques: une classification. Prepr., Univ. Genève, appeared in: Ann. Sc. Norm. Supér. Pisa, Cl. Sci., IV. Ser. 17, No. 2, 165–195 [1990]. Zb1.728.14041Google Scholar
  435. Wehlau, D. [1993a]: Equidimensional representations of 2-simple groups. J. Algebra 154, no. 2, 437–489MATHGoogle Scholar
  436. Wehlau, D. [1993b]: Equidimensional representations and associated cones [to appear in J. Algebra]Google Scholar
  437. Wehlau, D. [1993c]: Constructive invariant theory for tori [to appear in Ann. Inst. Fourier 43]Google Scholar
  438. Weil, A. [1955]: On algebraic groups of transformations. Am. J. Math. 77, 355–391. Zb1.65,142MATHGoogle Scholar
  439. Weisfeiler, B.Y. [1966]: On a class of unipotent subgroups of semisimple algebraic groups. Usp. Mat. Nauk 21, No. 2, 222–223 [Russian]Google Scholar
  440. Weitzenböck, R. [1923]: Invariantentheorie. Groningen: Noordhoff. 408 pp. FdM49,64Google Scholar
  441. Weitzenböck, R. [1932]: Über die Invarianten von linearen Gruppen. Acta Math. 58, 231–293MATHGoogle Scholar
  442. Weyl, H. [1925]: Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen. I. Math. Z. 23, 271–309. FdM51,319MATHGoogle Scholar
  443. Weyl, H. [1926]: Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen. II, III, IV. Math. Z. 24, 328–376, 377–395, 789–791. FdM51,319Google Scholar
  444. Weyl, H. [1939]: The Classical Groups. Their Invariants and Representations. Princeton, N.J.: Princeton University Press. 302 pp. Zb1.20,206Google Scholar
  445. Weyman, J. [1989]: The equations of conjugacy classes of nilpotent matrices. Invent. Math. 98, 229–245MATHGoogle Scholar
  446. Weyman, J. [1992]: Some open problems in invariant theory. In: Free Resolutions in Comm. Algebra and Algebraic Geometry [Eisenbud, D., Huneke, C., Eds.]. Jones & Bartlett, 139–146Google Scholar
  447. Young, A. [1933]: On quantitative substitutional analysis. VII. Proc. Lond. Math. Soc. 36, No. 4, 304–368. Zb1.8,49Google Scholar
  448. Zalesskiï, A.E. [1983]: The fixed algebra of a group generated by reflections is not always free. Arch. Math. 41, no. 5, 434–437Google Scholar
  449. Zariski, O., Samuel, P. [1960]: Commutative Algebra. Vol. II. Princeton, N.J.: D. van Nostrand. 414 pp. Zb1.121,278Google Scholar
  450. Zhelobenko, D.P. [1970]: Compact Lie Groups and Their Representations. Moscow: Nauka. 664 pp. English transi.: Transi. Math. Monogr. Vol. 40, Providence [1973]. Zb1.228.22013Google Scholar
  451. Séminaire C. Chevalley [1958]: Anneaux de Chow et Applications. Paris: École Normale Supérieure. 135 pp.Google Scholar
  452. Some open problems in invariant theory [1992]: Contemp. Math. 131, 485–497 [prepared for publication by Popov, V.L., and Vinberg, E.B.]Google Scholar

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© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • V. L. Popov
  • E. B. Vinberg

There are no affiliations available

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