Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Singularities in the nilpotent scheme of a classical group
HTML articles powered by AMS MathViewer

by Wim Hesselink PDF
Trans. Amer. Math. Soc. 222 (1976), 1-32 Request permission

Abstract:

If $(X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition ${\text {ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $N(G)$ of ${\text {Lie}}(G)$ is discussed. We prove that ${\text {ord}}(x,N(G)/k)$ characterizes the orbits in $N(G)$, their codimension and their adjacency structure, provided that G is $G{l_n}$, or $S{p_n}$ and $1/2 \in k$. For $S{O_n}$ only partial results are obtained. We give presentations of some singularities of $N(G)$. Tables for its orbit structure are added.
References
  • V. I. Arnol′d, On matrices depending on parameters, Uspehi Mat. Nauk 26 (1971), no. 2(158), 101–114 (Russian). MR 0301242
  • Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
  • N. Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1272, Hermann, Paris, 1959 (French). MR 0107661
  • —, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chaps. IV, V, VI, Actualités Sci. Indust., no. 1337, Hermann, Paris, 1968. MR 39 #1590.
  • E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798
  • C. Chevalley, Classification des groupes de Lie algébriques, Notes polycopiées, Inst. H. Poincaré, Paris, 1956-1958.
  • Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790
  • Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
  • Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. MR 132079, DOI 10.2307/1970336
  • Murray Gerstenhaber, Dominance over the classical groups, Ann. of Math. (2) 74 (1961), 532–569. MR 136683, DOI 10.2307/1970297
  • A. Grothendieck and J. A. Dieudonné, Éléments de géométrie-algébrique. I, Die Grundlehren der math. Wissenschaften, Band 166, Springer-Verlag, Berlin, 1971. —, Éléments de géométrie algébrique. II, III, IV, Inst. Hautes Études Sci. Publ. Math. Nos. 8, 11, 17, 20, 24, 28, 32 (1961-1967). MR 29 #1212; 30 #3885; 33 #7330; 36 #177b, c; #178; 39 #220. W. H. Hesselink, Singularities in the nilpotent scheme of a classical group, Thesis, Utrecht, 1975.
  • Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
  • Birger Iversen, Generic local structure of the morphisms in commutative algebra, Lecture Notes in Mathematics, Vol. 310, Springer-Verlag, Berlin-New York, 1973. MR 0360575, DOI 10.1007/BFb0060790
  • Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
  • David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
  • Jean-Jacques Risler, Algèbre symétrique d’un idéal, Sur Quelques Problèmes d’Algèbre, Université de Montpellier, Montpellier, 1969, pp. 61–69 (French). MR 0254040
  • T. A. Springer, The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 373–391. MR 0263830
  • T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
  • Robert Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, Vol. 366, Springer-Verlag, Berlin-New York, 1974. Notes by Vinay V. Deodhar. MR 0352279, DOI 10.1007/BFb0067854
  • F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic $p$, Ann. Sci. École Norm. Sup. (4) 5 (1972), 217–240. MR 308227, DOI 10.24033/asens.1225
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 14B05, 14L15
  • Retrieve articles in all journals with MSC: 14B05, 14L15
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 222 (1976), 1-32
  • MSC: Primary 14B05; Secondary 14L15
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0429875-8
  • MathSciNet review: 0429875