Skip to main content

Representations of Compact Groups

  • Chapter
Lie Groups

Part of the book series: Universitext ((UTX))

  • 3913 Accesses

Abstract

In this introduction, we give the basic definitions of representation theory, followed by a summary of the main results for compact (Lie) groups.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Van den Ban, E.P. (1997) Induced representations and the Langlands classification. in: Bailey, T.N., Knapp, A.W. (eds.) Proc. Sympos. Pure Math. 61 (1997) 123155. Amer. Math. Soc., Providence

    Google Scholar 

  • Bourbaki, N. (1965) Éléments de Mathématique. Topologie Générale, Chap. 1 et 2. Hermann, Paris

    Google Scholar 

  • Bourbaki, N. (1966) Éléments de Mathématique. Espaces Vectoriels Topologiques, Chap. 1 et 2. Hermann, Paris

    Google Scholar 

  • Bourbaki, N. (1964) Éléments de Mathématique. Espaces Vectoriels Topologiques, Chap. III-V. Hermann, Paris

    Google Scholar 

  • Bourbaki, N. (1975) Éléments de Mathématique. Groupes et Alg¨¨bres de Lie, Chap. 7 et 8. Hermann, Paris

    Google Scholar 

  • Bourbaki, N. (1982) Éléments de Mathématique. Groupes et Alg¨¨bres de Lie, Chap. 9. Masson, Paris

    Google Scholar 

  • Bröcker, T., tom Dieck, T. (1985) Repesentations of Compact Lie Groups. Springer-Verlag, New York Berlin Heidelberg Tokyo

    Google Scholar 

  • Burnside, W. (1904) On the reduction of a group of transformations of homogeneous linear substitutions of finite order. Acta Math. 28 (1904) 369-387

    Google Scholar 

  • Burnside, W. (1904) On the representation of a group of finite order as an irreducible group of linear substitutions and the direct establishment of the relations between the group characters. Proc. London Math. Soc. 1 (1904) 117-123

    Google Scholar 

  • Cartan, É. (1913) Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. France 41 (1913) 53-96 = OEuvres Compl¨¨tes, Partie I, pp. 355-398. Gauthier-Villars, Paris 1952

    Google Scholar 

  • Curtis, C.W., Reiner, I. (1981/1987) Methods of Representation Theory, Volumes

    Google Scholar 

  • I and II. John Wiley & Sons, New York Chichester Brisbane Toronto Demazure, M. (1974) Une nouvelle formule des caract¨res. Bull. Sc. Math. (2) 98(1974) 163-172

    Google Scholar 

  • Freudenthal, H., de Vries, H. (1969) Linear Lie Groups. Academic Press, New York London

    Google Scholar 

  • Frobenius, F.G. (1878) Uber lineare Substitutionen und bilineare Formen. J. reine angew. Math. 84 (1878) 1-63 = Gesammelte Abh., Bd. II, pp. 343-405. Springer-Verlag, Berlin Heidelberg New York 1968

    Google Scholar 

  • Frobenius, F.G. A series of articles in: Sitzungsber. Königl. Preuss. Akad. Wiss., Phys.-Math. Kl. = Gesammelte Abh., Bd. III. Springer-Verlag, Berlin Heidelberg New York 1968:

    Google Scholar 

  • (a) Uber Gruppencharaktere. 1896, pp. 985-1021 = pp. 1-37;

    Google Scholar 

  • (b) Uber die Primfactoren der Gruppendeterminante. 1896, pp. 1343-1382 = pp. 38-77;

    Google Scholar 

  • (c) Über die Darstellung der endlichen Gruppen durch lineare Substitutionen. 1897, pp. 994-1015 = pp. 82-103;

    Google Scholar 

  • (d) Uber Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen. 1898, pp. 501-515 = pp. 104-118;

    Google Scholar 

  • (e) Uber die Composition der Charaktere einer Gruppe. 1899, pp. 330-339 = pp. 119-147

    Google Scholar 

  • Frobenius, F.G., Schur, I. (1906) Uber die reellen Darstellungen der endlichen Gruppen. Sitzungsber. Königl. Preuss. Akad. Wiss., Phys.-Math. Kl. 1906, pp. 186-208 = Gesammelte Abh., Band. III, pp. 355-377. Springer-Verlag, Berlin Heidelberg New York 1968

    Google Scholar 

  • Grauert, H. (1958) On Levi's problem and the imbedding of real-analytic manifolds. Ann. of Math. (2) 68 (1958) 460-472

    Google Scholar 

  • Griffiths, P., Harris, J. (1978) Principles of Algebraic Geometry. J.Wiley and Sons, New York Chichester Brisbane Toronto

    MATH  Google Scholar 

  • Haar, A. (1933) Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. of Math. (II) 34 (1933) 147-169

    Google Scholar 

  • Harish-Chandra (1951) On some applications of the universal enveloping algebra of a semi-simple Lie algebra. Trans. Amer. Math. Soc. 70 (1951) 28-95 = Collected Papers, Vol. I, pp. 292-360. Springer-Verlag, New York Berlin Heidelberg Tokyo 1984

    Google Scholar 

  • Harish-Chandra (1956) On a lemma of F. Bruhat. J. Math. Pures Appl. (9) 35 (1956) 203-210 = Collected Papers, Vol. II, pp. 223-230. Springer-Verlag, New York Berlin Heidelberg Tokyo 1984

    Google Scholar 

  • Harish-Chandra (1967) Characters of semi-simple Lie groups. Symposia on Theoretical Physics 4 (1967) 137-142 = Collected Papers, Vol. III, pp. 655-660. Springer-Verlag, New York Berlin Heidelberg Tokyo 1984

    Google Scholar 

  • Hawkins, T. (1978) The creation of the theory of group characters. In: History of Analysis. Rice Univ. Studies 64 (1978) 57-71

    Google Scholar 

  • Helgason, S. (1962) Differential Geometry and Symmetric Spaces. Academic Press, New York London

    MATH  Google Scholar 

  • Hirsch, M.W., Smale, S. (1974) Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York San Francisco London

    MATH  Google Scholar 

  • Hörmander, L. (1983) The Analysis of Linear Partial Differential Operators, Vol.

    Google Scholar 

  • I. Springer-Verlag, Berlin Heidelberg New York Tokyo

    Google Scholar 

  • Hurwitz, A. (1897) Uber die Erzeugung der Invarianten durch Integration. Göttinger Nachrichten 1897, pp. 71-90 = Mathematische Werke, Band II, pp. 546-564. Birkhäuser, Basel 1933

    Google Scholar 

  • Iwahori, N. (1959) On real irreducible representations of Lie algebras. Nagoya Math. J. 14 (1959) 59-83

    Google Scholar 

  • Kostant, B (1959) A formula for the multiplicity of a weight. Trans. Amer. Math. Soc. 93 (1959) 53-73

    Google Scholar 

  • Kostant, B. (1961) Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961) 329-387

    Google Scholar 

  • Maschke, H. (1899) Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehend verschwindende Coefficienten auftreten, intransitiv sind. Math. Ann. 52 (1899) 363-368

    Google Scholar 

  • Molien, T. (1893) Uber Systeme höheren complexer Zahlen. Math. Ann. 41 (1893) 83-156; Berichtigung, ibidem 42 (1893) 308-312

    Google Scholar 

  • Molien, T. (1897) Eine Bemerkung zur Theorie der homogenen Substitutionsgruppen. Sitzungsber. d. Naturforscher-Ges. Dorpat 18 (1897) p. 259. See also: Uber die Invarianten der linearen Substitutionsgruppen. Sitzungsber. Königl. Preuss. Akad. Wiss., Phys.-Math. Kl. 1897, pp. 1152-1156.

    Google Scholar 

  • Moore, E.H. (1898) An universal invariant for finite groups of linear substitutions: with applications in the theory of the canonical form of a linear substitution of finite period. Math. Ann. 50 (1898) 213-219

    Google Scholar 

  • von Neumann, J. (1936) The uniqueness of Haar's measure. Mat. Sbornik 1 (43) (1936) 721-734 = Collected Works, Vol. IV, pp. 91-104. Pergamon Press, Oxford 1962

    Google Scholar 

  • von Neumann, J. (1934) Almost periodic functions in a group I. Trans. Amer. Math. Soc. 36 (1934) 445-492 = Collected Works, Vol. II, pp. 454-501. Pergamon Press, Oxford 1961

    Google Scholar 

  • Peter, F., Weyl, H. (1927) Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe. Math. Ann. 97 (1927) 737-755 = Gesammelte Abh., Bd. III, pp. 58-75. Springer-Verlag, Berlin Heidelberg New York 1968

    Google Scholar 

  • Schmid, W. (1985) Recent developments in representation theory. In: Hirzebruch, F., Schwermer, J., Suter, S. (eds.) Arbeitstagung Bonn 1984. Lecture Notes in Math. 1111, pp, 135-153. Springer-Verlag, Berlin Heidelberg New York Tokyo

    Chapter  Google Scholar 

  • Schur, I. (1901) Uber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. Diss., Berlin 1901 = Gesammelte Abh., Bd. I, pp. 1-72. Springer-Verlag, Berlin Heidelberg New York 1973

    Google Scholar 

  • Schur, I. (1905) Neue Begründung der Theorie der Gruppencharakteren. Sitzungsber. Königl. Preuss. Akad. Wiss., Phys.-Math. Kl. 1905, pp. 406-432 = Gesammelte Abh., Bd. I, pp. 143-169. Springer-Verlag, Berlin Heidelberg New York 1973

    Google Scholar 

  • Schur, I. (1924) Neue Anwendung der Integralrechnung auf Probleme der Invariantentheorie. Sitzungsber. Preuss. Akad. Wiss., Phys.- Math. Kl. 1924 I. Mitteilung, pp. 189-208; II. Über die Darstellung der Drehungsgruppe durch lineare homogene Substitution. pp. 297-321; III. Vereinfachung des Integralkalküls. Realitätsfragen. pp. 346-355 = Gesammelte Abh., Bd. II, pp. 440-494. Springer-Verlag, Berlin Heidelberg New York 1973

    Google Scholar 

  • Serre, J-P. (1954) Représentations linéaires et espaces homog¨¨nes Kählériens des

    Google Scholar 

  • groupes de Lie compacts. Séminaire Bourbaki, Exp. No. 100 (1954) 1-8 Speiser, A. (1927) Theorie der Gruppen von endlicher Ordnung. 2er Auflage. Julius Springer, Berlin

    Google Scholar 

  • Steinberg, R. (1961) A general Clebsch-Gordan theorem. Bull. Amer. Math. Soc. 67 (1961) 406-407

    Article  MathSciNet  MATH  Google Scholar 

  • Tits, J. (1955) Sur certaines classes d'espaces homog¨¨nes de groupes de Lie. Acad. Roy. Belg. Cl. Sci. Mém. Coll. 29, No. 3

    Google Scholar 

  • Treves, F. (1967) xxxxTopological Vector Spaces, Distributions and Kernels Academic Press, New York London

    Google Scholar 

  • Vogan Jr., D.A. (1997-I) Cohomology and group representations. in: Bailey, T.N., Knapp, A.W. (eds.) Proc. Sympos. Pure Math. 61 (1997) 219-243. Amer. Math. Soc., Providence

  • Vogan Jr., D.A. (1997-II) The orbit method and unitary representations for reductive Lie groups. In: (rsted, B., Schlichtkrull, H. (eds.) Algebraic and Analytic Methods in Representation Theory. 1997, pp. 243-339. Academic Press, San Diego London Boston New York Sydney Tokyo Toronto

    Google Scholar 

  • van der Waerden, B.L. (1966) Algebra I. Springer-Verlag, Berlin Heidelberg New York

    Google Scholar 

  • Wedderburn, J.H.M. (1908) On hypercomplex numbers. Proc. London Math. Soc. (2) 6 (1908) 77-118

    Article  MATH  Google Scholar 

  • Weil, A. (1940) l'Intégration dans les Groupes Topologiques. Hermann, Paris

    Google Scholar 

  • Weyl, H. (1925/26) Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I, II, III, Nachtrag. Math. Z. 23 (1925) 271-309; 24 (1926) 328-376, 377-395, 789-791 = Gesammelte Abh., Band II, pp. 543-647. Springer-Verlag, Berlin Heidelberg New York 1968

    Google Scholar 

  • Walter, W. (1986) Gewöhnliche Differentialgleichungen. 3er Aufl. Springer-Verlag, Berlin Heidelberg New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Duistermaat, J.J., Kolk, J.A.C. (2000). Representations of Compact Groups. In: Lie Groups. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56936-4_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56936-4_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15293-4

  • Online ISBN: 978-3-642-56936-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics